cdlemk26-3

Description: Part of proof of Lemma K of Crawley p. 118. Eliminate the x requirements from cdlemk25-3 . (Contributed by NM, 10-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	cdlemk26-3	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D))) \to (D Y G) (P) = (C Y G) (P)$

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		simp11l	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D))) \to K \in HL$
12		simp11r	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D))) \to W \in H$
13	1 6 7 8	cdlemftr3	$⊢ (K \in HL \land W \in H) \to \exists x \in T (x \neq {I ↾}_{B} \land (R (x) \neq R (F) \land R (x) \neq R (G) \land R (x) \neq R (D)))$
14	11 12 13	syl2anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D))) \to \exists x \in T (x \neq {I ↾}_{B} \land (R (x) \neq R (F) \land R (x) \neq R (G) \land R (x) \neq R (D)))$
15		simp111	$⊢ ((((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D))) \land x \in T \land (x \neq {I ↾}_{B} \land (R (x) \neq R (F) \land R (x) \neq R (G) \land R (x) \neq R (D)))) \to (K \in HL \land W \in H)$
16		simp112	$⊢ ((((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D))) \land x \in T \land (x \neq {I ↾}_{B} \land (R (x) \neq R (F) \land R (x) \neq R (G) \land R (x) \neq R (D)))) \to (F \in T \land D \in T \land N \in T)$
17		simp13l	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D))) \to G \in T$
18	17	3ad2ant1	$⊢ ((((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D))) \land x \in T \land (x \neq {I ↾}_{B} \land (R (x) \neq R (F) \land R (x) \neq R (G) \land R (x) \neq R (D)))) \to G \in T$
19		simp13r	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D))) \to C \in T$
20	19	3ad2ant1	$⊢ ((((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D))) \land x \in T \land (x \neq {I ↾}_{B} \land (R (x) \neq R (F) \land R (x) \neq R (G) \land R (x) \neq R (D)))) \to C \in T$
21		simp2	$⊢ ((((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D))) \land x \in T \land (x \neq {I ↾}_{B} \land (R (x) \neq R (F) \land R (x) \neq R (G) \land R (x) \neq R (D)))) \to x \in T$
22	18 20 21	3jca	$⊢ ((((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D))) \land x \in T \land (x \neq {I ↾}_{B} \land (R (x) \neq R (F) \land R (x) \neq R (G) \land R (x) \neq R (D)))) \to (G \in T \land C \in T \land x \in T)$
23		simp121	$⊢ ((((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D))) \land x \in T \land (x \neq {I ↾}_{B} \land (R (x) \neq R (F) \land R (x) \neq R (G) \land R (x) \neq R (D)))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
24		simp122	$⊢ ((((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D))) \land x \in T \land (x \neq {I ↾}_{B} \land (R (x) \neq R (F) \land R (x) \neq R (G) \land R (x) \neq R (D)))) \to (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B})$
25		simp23l	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D))) \to G \neq {I ↾}_{B}$
26	25	3ad2ant1	$⊢ ((((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D))) \land x \in T \land (x \neq {I ↾}_{B} \land (R (x) \neq R (F) \land R (x) \neq R (G) \land R (x) \neq R (D)))) \to G \neq {I ↾}_{B}$
27		simp23r	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D))) \to C \neq {I ↾}_{B}$
28	27	3ad2ant1	$⊢ ((((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D))) \land x \in T \land (x \neq {I ↾}_{B} \land (R (x) \neq R (F) \land R (x) \neq R (G) \land R (x) \neq R (D)))) \to C \neq {I ↾}_{B}$
29		simp3l	$⊢ ((((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D))) \land x \in T \land (x \neq {I ↾}_{B} \land (R (x) \neq R (F) \land R (x) \neq R (G) \land R (x) \neq R (D)))) \to x \neq {I ↾}_{B}$
30	26 28 29	3jca	$⊢ ((((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D))) \land x \in T \land (x \neq {I ↾}_{B} \land (R (x) \neq R (F) \land R (x) \neq R (G) \land R (x) \neq R (D)))) \to (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B} \land x \neq {I ↾}_{B})$
31		simp13l	$⊢ ((((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D))) \land x \in T \land (x \neq {I ↾}_{B} \land (R (x) \neq R (F) \land R (x) \neq R (G) \land R (x) \neq R (D)))) \to (R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F))$
32		simp13r	$⊢ ((((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D))) \land x \in T \land (x \neq {I ↾}_{B} \land (R (x) \neq R (F) \land R (x) \neq R (G) \land R (x) \neq R (D)))) \to R (G) \neq R (D)$
33		simp3r3	$⊢ ((((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D))) \land x \in T \land (x \neq {I ↾}_{B} \land (R (x) \neq R (F) \land R (x) \neq R (G) \land R (x) \neq R (D)))) \to R (x) \neq R (D)$
34		simp3r1	$⊢ ((((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D))) \land x \in T \land (x \neq {I ↾}_{B} \land (R (x) \neq R (F) \land R (x) \neq R (G) \land R (x) \neq R (D)))) \to R (x) \neq R (F)$
35		simp3r2	$⊢ ((((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D))) \land x \in T \land (x \neq {I ↾}_{B} \land (R (x) \neq R (F) \land R (x) \neq R (G) \land R (x) \neq R (D)))) \to R (x) \neq R (G)$
36	35	necomd	$⊢ ((((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D))) \land x \in T \land (x \neq {I ↾}_{B} \land (R (x) \neq R (F) \land R (x) \neq R (G) \land R (x) \neq R (D)))) \to R (G) \neq R (x)$
37	33 34 36	3jca	$⊢ ((((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D))) \land x \in T \land (x \neq {I ↾}_{B} \land (R (x) \neq R (F) \land R (x) \neq R (G) \land R (x) \neq R (D)))) \to (R (x) \neq R (D) \land R (x) \neq R (F) \land R (G) \neq R (x))$
38	1 2 3 4 5 6 7 8 9 10	cdlemk25-3	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T \land x \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B} \land x \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D) \land (R (x) \neq R (D) \land R (x) \neq R (F) \land R (G) \neq R (x)))) \to (D Y G) (P) = (C Y G) (P)$
39	15 16 22 23 24 30 31 32 37 38	syl333anc	$⊢ ((((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D))) \land x \in T \land (x \neq {I ↾}_{B} \land (R (x) \neq R (F) \land R (x) \neq R (G) \land R (x) \neq R (D)))) \to (D Y G) (P) = (C Y G) (P)$
40	39	rexlimdv3a	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D))) \to (\exists x \in T (x \neq {I ↾}_{B} \land (R (x) \neq R (F) \land R (x) \neq R (G) \land R (x) \neq R (D))) \to (D Y G) (P) = (C Y G) (P))$
41	14 40	mpd	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D))) \to (D Y G) (P) = (C Y G) (P)$

Metamath Proof Explorer

Theorem cdlemk26-3

Proof