cdlemk28-3

Description: Part of proof of Lemma K of Crawley p. 118. (Contributed by NM, 14-Jul-2013)

	Ref	Expression
Hypotheses	cdlemk3.b	$⊢ B = {Base}_{K}$
	cdlemk3.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdlemk3.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdlemk3.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdlemk3.a	$⊢ A = Atoms (K)$
	cdlemk3.h	$⊢ H = LHyp (K)$
	cdlemk3.t	$⊢ T = LTrn (K) (W)$
	cdlemk3.r	$⊢ R = trL (K) (W)$
	cdlemk3.s	$⊢ S = (f \in T ⟼ (ι i \in T \| i (P) = (P \underset{˙}{\lor} R (f)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (f \circ F^{-1}))))$
	cdlemk3.u1	$⊢ Y = (d \in T, e \in T ⟼ (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (e)) \underset{˙}{\land} (S (d) (P) \underset{˙}{\lor} R (e \circ d^{-1}))))$
Assertion	cdlemk28-3	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to \exists z \in T \forall b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \to z = b Y G)$

Proof

Step	Hyp	Ref	Expression
1		cdlemk3.b	$⊢ B = {Base}_{K}$
2		cdlemk3.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		cdlemk3.j	$⊢ \underset{˙}{\lor} = join (K)$
4		cdlemk3.m	$⊢ \underset{˙}{\land} = meet (K)$
5		cdlemk3.a	$⊢ A = Atoms (K)$
6		cdlemk3.h	$⊢ H = LHyp (K)$
7		cdlemk3.t	$⊢ T = LTrn (K) (W)$
8		cdlemk3.r	$⊢ R = trL (K) (W)$
9		cdlemk3.s	$⊢ S = (f \in T ⟼ (ι i \in T \| i (P) = (P \underset{˙}{\lor} R (f)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (f \circ F^{-1}))))$
10		cdlemk3.u1	$⊢ Y = (d \in T, e \in T ⟼ (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (e)) \underset{˙}{\land} (S (d) (P) \underset{˙}{\lor} R (e \circ d^{-1}))))$
11		simp1	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to (K \in HL \land W \in H)$
12		simp21l	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to F \in T$
13		simp21r	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to F \neq {I ↾}_{B}$
14		simp23	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to N \in T$
15	12 13 14	3jca	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to (F \in T \land F \neq {I ↾}_{B} \land N \in T)$
16		simp22l	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to G \in T$
17		simp22r	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to G \neq {I ↾}_{B}$
18		simp3r	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to R (F) = R (N)$
19	16 17 18	3jca	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to (G \in T \land G \neq {I ↾}_{B} \land R (F) = R (N))$
20		simp3l	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
21	1 2 3 4 5 6 7 8 9 10	cdlemk26b-3	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B} \land R (F) = R (N))) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to \exists b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land b Y G \in T)$
22	11 15 19 20 21	syl31anc	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to \exists b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land b Y G \in T)$
23		simp11	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (b \in T \land a \in T) \land ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (a \neq {I ↾}_{B} \land R (a) \neq R (F) \land R (a) \neq R (G)))) \to (K \in HL \land W \in H)$
24	12	3ad2ant1	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (b \in T \land a \in T) \land ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (a \neq {I ↾}_{B} \land R (a) \neq R (F) \land R (a) \neq R (G)))) \to F \in T$
25		simp2l	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (b \in T \land a \in T) \land ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (a \neq {I ↾}_{B} \land R (a) \neq R (F) \land R (a) \neq R (G)))) \to b \in T$
26		simp123	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (b \in T \land a \in T) \land ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (a \neq {I ↾}_{B} \land R (a) \neq R (F) \land R (a) \neq R (G)))) \to N \in T$
27	24 25 26	3jca	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (b \in T \land a \in T) \land ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (a \neq {I ↾}_{B} \land R (a) \neq R (F) \land R (a) \neq R (G)))) \to (F \in T \land b \in T \land N \in T)$
28	16	3ad2ant1	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (b \in T \land a \in T) \land ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (a \neq {I ↾}_{B} \land R (a) \neq R (F) \land R (a) \neq R (G)))) \to G \in T$
29		simp2r	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (b \in T \land a \in T) \land ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (a \neq {I ↾}_{B} \land R (a) \neq R (F) \land R (a) \neq R (G)))) \to a \in T$
30	28 29	jca	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (b \in T \land a \in T) \land ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (a \neq {I ↾}_{B} \land R (a) \neq R (F) \land R (a) \neq R (G)))) \to (G \in T \land a \in T)$
31		simp13l	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (b \in T \land a \in T) \land ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (a \neq {I ↾}_{B} \land R (a) \neq R (F) \land R (a) \neq R (G)))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
32		simp13r	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (b \in T \land a \in T) \land ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (a \neq {I ↾}_{B} \land R (a) \neq R (F) \land R (a) \neq R (G)))) \to R (F) = R (N)$
33	13	3ad2ant1	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (b \in T \land a \in T) \land ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (a \neq {I ↾}_{B} \land R (a) \neq R (F) \land R (a) \neq R (G)))) \to F \neq {I ↾}_{B}$
34		simp3l1	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (b \in T \land a \in T) \land ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (a \neq {I ↾}_{B} \land R (a) \neq R (F) \land R (a) \neq R (G)))) \to b \neq {I ↾}_{B}$
35	32 33 34	3jca	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (b \in T \land a \in T) \land ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (a \neq {I ↾}_{B} \land R (a) \neq R (F) \land R (a) \neq R (G)))) \to (R (F) = R (N) \land F \neq {I ↾}_{B} \land b \neq {I ↾}_{B})$
36	17	3ad2ant1	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (b \in T \land a \in T) \land ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (a \neq {I ↾}_{B} \land R (a) \neq R (F) \land R (a) \neq R (G)))) \to G \neq {I ↾}_{B}$
37		simp3r1	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (b \in T \land a \in T) \land ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (a \neq {I ↾}_{B} \land R (a) \neq R (F) \land R (a) \neq R (G)))) \to a \neq {I ↾}_{B}$
38	36 37	jca	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (b \in T \land a \in T) \land ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (a \neq {I ↾}_{B} \land R (a) \neq R (F) \land R (a) \neq R (G)))) \to (G \neq {I ↾}_{B} \land a \neq {I ↾}_{B})$
39		simp3r3	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (b \in T \land a \in T) \land ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (a \neq {I ↾}_{B} \land R (a) \neq R (F) \land R (a) \neq R (G)))) \to R (a) \neq R (G)$
40	39	necomd	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (b \in T \land a \in T) \land ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (a \neq {I ↾}_{B} \land R (a) \neq R (F) \land R (a) \neq R (G)))) \to R (G) \neq R (a)$
41		simp3r2	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (b \in T \land a \in T) \land ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (a \neq {I ↾}_{B} \land R (a) \neq R (F) \land R (a) \neq R (G)))) \to R (a) \neq R (F)$
42		simp3l2	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (b \in T \land a \in T) \land ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (a \neq {I ↾}_{B} \land R (a) \neq R (F) \land R (a) \neq R (G)))) \to R (b) \neq R (F)$
43	40 41 42	3jca	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (b \in T \land a \in T) \land ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (a \neq {I ↾}_{B} \land R (a) \neq R (F) \land R (a) \neq R (G)))) \to (R (G) \neq R (a) \land R (a) \neq R (F) \land R (b) \neq R (F))$
44		simp3l3	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (b \in T \land a \in T) \land ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (a \neq {I ↾}_{B} \land R (a) \neq R (F) \land R (a) \neq R (G)))) \to R (b) \neq R (G)$
45	44	necomd	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (b \in T \land a \in T) \land ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (a \neq {I ↾}_{B} \land R (a) \neq R (F) \land R (a) \neq R (G)))) \to R (G) \neq R (b)$
46	1 2 3 4 5 6 7 8 9 10	cdlemk27-3	$⊢ (((K \in HL \land W \in H) \land (F \in T \land b \in T \land N \in T) \land (G \in T \land a \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land b \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land a \neq {I ↾}_{B})) \land ((R (G) \neq R (a) \land R (a) \neq R (F) \land R (b) \neq R (F)) \land R (G) \neq R (b))) \to b Y G = a Y G$
47	23 27 30 31 35 38 43 45 46	syl332anc	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (b \in T \land a \in T) \land ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (a \neq {I ↾}_{B} \land R (a) \neq R (F) \land R (a) \neq R (G)))) \to b Y G = a Y G$
48	47	3exp	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to ((b \in T \land a \in T) \to (((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (a \neq {I ↾}_{B} \land R (a) \neq R (F) \land R (a) \neq R (G))) \to b Y G = a Y G))$
49	48	ralrimivv	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to \forall b \in T \forall a \in T (((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (a \neq {I ↾}_{B} \land R (a) \neq R (F) \land R (a) \neq R (G))) \to b Y G = a Y G)$
50		neeq1	$⊢ b = a \to (b \neq {I ↾}_{B} \leftrightarrow a \neq {I ↾}_{B})$
51		fveq2	$⊢ b = a \to R (b) = R (a)$
52	51	neeq1d	$⊢ b = a \to (R (b) \neq R (F) \leftrightarrow R (a) \neq R (F))$
53	51	neeq1d	$⊢ b = a \to (R (b) \neq R (G) \leftrightarrow R (a) \neq R (G))$
54	50 52 53	3anbi123d	$⊢ b = a \to ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \leftrightarrow (a \neq {I ↾}_{B} \land R (a) \neq R (F) \land R (a) \neq R (G)))$
55		oveq1	$⊢ b = a \to b Y G = a Y G$
56	54 55	reusv3	$⊢ \exists b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land b Y G \in T) \to (\forall b \in T \forall a \in T (((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (a \neq {I ↾}_{B} \land R (a) \neq R (F) \land R (a) \neq R (G))) \to b Y G = a Y G) \leftrightarrow \exists z \in T \forall b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \to z = b Y G))$
57	56	biimpd	$⊢ \exists b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land b Y G \in T) \to (\forall b \in T \forall a \in T (((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (a \neq {I ↾}_{B} \land R (a) \neq R (F) \land R (a) \neq R (G))) \to b Y G = a Y G) \to \exists z \in T \forall b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \to z = b Y G))$
58	22 49 57	sylc	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to \exists z \in T \forall b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \to z = b Y G)$

Metamath Proof Explorer

Theorem cdlemk28-3

Proof