cdleme28b

Description: Lemma for cdleme25b . TODO: FIX COMMENT. (Contributed by NM, 6-Feb-2013)

	Ref	Expression
Hypotheses	cdleme26.b	$⊢ B = {Base}_{K}$
	cdleme26.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdleme26.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdleme26.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdleme26.a	$⊢ A = Atoms (K)$
	cdleme26.h	$⊢ H = LHyp (K)$
	cdleme27.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
	cdleme27.f	$⊢ F = (s \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} s) \underset{˙}{\land} W))$
	cdleme27.z	$⊢ Z = (z \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} z) \underset{˙}{\land} W))$
	cdleme27.n	$⊢ N = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (Z \underset{˙}{\lor} ((s \underset{˙}{\lor} z) \underset{˙}{\land} W))$
	cdleme27.d	$⊢ D = (ι u \in B \| \forall z \in A ((\neg z \underset{˙}{\leq} W \land \neg z \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to u = N))$
	cdleme27.c	$⊢ C = if (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q), D, F)$
	cdleme27.g	$⊢ G = (t \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} t) \underset{˙}{\land} W))$
	cdleme27.o	$⊢ O = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (Z \underset{˙}{\lor} ((t \underset{˙}{\lor} z) \underset{˙}{\land} W))$
	cdleme27.e	$⊢ E = (ι u \in B \| \forall z \in A ((\neg z \underset{˙}{\leq} W \land \neg z \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to u = O))$
	cdleme27.y	$⊢ Y = if (t \underset{˙}{\leq} (P \underset{˙}{\lor} Q), E, G)$
Assertion	cdleme28b	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to C \underset{˙}{\lor} (X \underset{˙}{\land} W) = Y \underset{˙}{\lor} (X \underset{˙}{\land} W)$

Proof

Step	Hyp	Ref	Expression
1		cdleme26.b	$⊢ B = {Base}_{K}$
2		cdleme26.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		cdleme26.j	$⊢ \underset{˙}{\lor} = join (K)$
4		cdleme26.m	$⊢ \underset{˙}{\land} = meet (K)$
5		cdleme26.a	$⊢ A = Atoms (K)$
6		cdleme26.h	$⊢ H = LHyp (K)$
7		cdleme27.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
8		cdleme27.f	$⊢ F = (s \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} s) \underset{˙}{\land} W))$
9		cdleme27.z	$⊢ Z = (z \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} z) \underset{˙}{\land} W))$
10		cdleme27.n	$⊢ N = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (Z \underset{˙}{\lor} ((s \underset{˙}{\lor} z) \underset{˙}{\land} W))$
11		cdleme27.d	$⊢ D = (ι u \in B \| \forall z \in A ((\neg z \underset{˙}{\leq} W \land \neg z \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to u = N))$
12		cdleme27.c	$⊢ C = if (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q), D, F)$
13		cdleme27.g	$⊢ G = (t \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} t) \underset{˙}{\land} W))$
14		cdleme27.o	$⊢ O = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (Z \underset{˙}{\lor} ((t \underset{˙}{\lor} z) \underset{˙}{\land} W))$
15		cdleme27.e	$⊢ E = (ι u \in B \| \forall z \in A ((\neg z \underset{˙}{\leq} W \land \neg z \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to u = O))$
16		cdleme27.y	$⊢ Y = if (t \underset{˙}{\leq} (P \underset{˙}{\lor} Q), E, G)$
17		simp11l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to K \in HL$
18	17	hllatd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to K \in Lat$
19		simp11r	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to W \in H$
20		simp12	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
21		simp13	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
22		simp22	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to (s \in A \land \neg s \underset{˙}{\leq} W)$
23		simp21	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to P \neq Q$
24	1 2 3 4 5 6 7 8 9 10 11 12	cdleme27cl	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land P \neq Q)) \to C \in B$
25	17 19 20 21 22 23 24	syl222anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to C \in B$
26		simp33l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to X \in B$
27	1 6	lhpbase	$⊢ W \in H \to W \in B$
28	19 27	syl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to W \in B$
29	1 4	latmcl	$⊢ (K \in Lat \land X \in B \land W \in B) \to X \underset{˙}{\land} W \in B$
30	18 26 28 29	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to X \underset{˙}{\land} W \in B$
31	1 3	latjcl	$⊢ (K \in Lat \land C \in B \land X \underset{˙}{\land} W \in B) \to C \underset{˙}{\lor} (X \underset{˙}{\land} W) \in B$
32	18 25 30 31	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to C \underset{˙}{\lor} (X \underset{˙}{\land} W) \in B$
33		simp23	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to (t \in A \land \neg t \underset{˙}{\leq} W)$
34	1 2 3 4 5 6 7 13 9 14 15 16	cdleme27cl	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((t \in A \land \neg t \underset{˙}{\leq} W) \land P \neq Q)) \to Y \in B$
35	17 19 20 21 33 23 34	syl222anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to Y \in B$
36	1 3	latjcl	$⊢ (K \in Lat \land Y \in B \land X \underset{˙}{\land} W \in B) \to Y \underset{˙}{\lor} (X \underset{˙}{\land} W) \in B$
37	18 35 30 36	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to Y \underset{˙}{\lor} (X \underset{˙}{\land} W) \in B$
38		eqid	$⊢ (s \underset{˙}{\lor} t) \underset{˙}{\land} (X \underset{˙}{\land} W) = (s \underset{˙}{\lor} t) \underset{˙}{\land} (X \underset{˙}{\land} W)$
39	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 38	cdleme28a	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to (C \underset{˙}{\lor} (X \underset{˙}{\land} W)) \underset{˙}{\leq} (Y \underset{˙}{\lor} (X \underset{˙}{\land} W))$
40		simp11	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to (K \in HL \land W \in H)$
41		simp31	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to s \neq t$
42	41	necomd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to t \neq s$
43		simp32	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)$
44	43	ancomd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to (t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)$
45		simp33	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to (X \in B \land \neg X \underset{˙}{\leq} W)$
46		eqid	$⊢ (t \underset{˙}{\lor} s) \underset{˙}{\land} (X \underset{˙}{\land} W) = (t \underset{˙}{\lor} s) \underset{˙}{\land} (X \underset{˙}{\land} W)$
47	1 2 3 4 5 6 7 13 9 14 15 16 8 10 11 12 46	cdleme28a	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (t \in A \land \neg t \underset{˙}{\leq} W) \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land (t \neq s \land (t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to (Y \underset{˙}{\lor} (X \underset{˙}{\land} W)) \underset{˙}{\leq} (C \underset{˙}{\lor} (X \underset{˙}{\land} W))$
48	40 20 21 23 33 22 42 44 45 47	syl333anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to (Y \underset{˙}{\lor} (X \underset{˙}{\land} W)) \underset{˙}{\leq} (C \underset{˙}{\lor} (X \underset{˙}{\land} W))$
49	1 2 18 32 37 39 48	latasymd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to C \underset{˙}{\lor} (X \underset{˙}{\land} W) = Y \underset{˙}{\lor} (X \underset{˙}{\land} W)$

Metamath Proof Explorer

Theorem cdleme28b

Proof