Theorem cdleme31snd

Description: Part of proof of Lemma D in Crawley p. 113. (Contributed by NM, 1-Apr-2013)

	Ref	Expression
Hypotheses	cdleme31snd.d	$$⊢D=\left(t\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme31snd.n	$$⊢N=\left(v\underset{\dot{}}{\vee }V\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }v\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme31snd.e	$$⊢E=\left(O\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }O\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme31snd.o	$$⊢O=\left(S\underset{\dot{}}{\vee }V\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\right)$$
Assertion	cdleme31snd	$$⊢S\in A\to ⦋S/v⦌⦋N/t⦌D=E$$

Proof

Step	Hyp	Ref	Expression
1		cdleme31snd.d	$$⊢D=\left(t\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
2		cdleme31snd.n	$$⊢N=\left(v\underset{\dot{}}{\vee }V\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }v\right)\underset{\dot{}}{\wedge }W\right)\right)$$
3		cdleme31snd.e	$$⊢E=\left(O\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }O\right)\underset{\dot{}}{\wedge }W\right)\right)$$
4		cdleme31snd.o	$$⊢O=\left(S\underset{\dot{}}{\vee }V\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\right)$$
5		csbnestgw	$$⊢S\in A\to ⦋S/v⦌⦋N/t⦌D=⦋⦋S/v⦌N/t⦌D$$
6	2 4	cdleme31sc	$$⊢S\in A\to ⦋S/v⦌N=O$$
7	6	csbeq1d	$$⊢S\in A\to ⦋⦋S/v⦌N/t⦌D=⦋O/t⦌D$$
8	4	ovexi	$$⊢O\in \mathrm{V}$$
9	1 3	cdleme31sc	$$⊢O\in \mathrm{V}\to ⦋O/t⦌D=E$$
10	8 9	ax-mp	$$⊢⦋O/t⦌D=E$$
11	7 10	eqtrdi	$$⊢S\in A\to ⦋⦋S/v⦌N/t⦌D=E$$
12	5 11	eqtrd	$$⊢S\in A\to ⦋S/v⦌⦋N/t⦌D=E$$