Theorem cdleme31sc 40830

Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 31-Mar-2013.)

Hypotheses

Ref	Expression
cdleme31sc.c	⊢ 𝐶 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊)))
cdleme31sc.x	⊢ 𝑋 = ((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊)))

Assertion

Ref	Expression
cdleme31sc	⊢ (𝑅 ∈ 𝐴 → ⦋𝑅 / 𝑠⦌𝐶 = 𝑋)

Distinct variable groups:   𝐴,𝑠   ∨ ,𝑠   ∧ ,𝑠   𝑃,𝑠   𝑄,𝑠   𝑅,𝑠   𝑈,𝑠   𝑊,𝑠

Allowed substitution hints:   𝐶(𝑠)   𝑋(𝑠)

Proof of Theorem cdleme31sc

Step	Hyp	Ref	Expression
1		nfcvd 2899	. . 3 ⊢ (𝑅 ∈ 𝐴 → Ⅎ𝑠((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊))))
2		oveq1 7374	. . . 4 ⊢ (𝑠 = 𝑅 → (𝑠 ∨ 𝑈) = (𝑅 ∨ 𝑈))
3		oveq2 7375	. . . . . 6 ⊢ (𝑠 = 𝑅 → (𝑃 ∨ 𝑠) = (𝑃 ∨ 𝑅))
4	3	oveq1d 7382	. . . . 5 ⊢ (𝑠 = 𝑅 → ((𝑃 ∨ 𝑠) ∧ 𝑊) = ((𝑃 ∨ 𝑅) ∧ 𝑊))
5	4	oveq2d 7383	. . . 4 ⊢ (𝑠 = 𝑅 → (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊)) = (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊)))
6	2, 5	oveq12d 7385	. . 3 ⊢ (𝑠 = 𝑅 → ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊))) = ((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊))))
7	1, 6	csbiegf 3870	. 2 ⊢ (𝑅 ∈ 𝐴 → ⦋𝑅 / 𝑠⦌((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊))) = ((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊))))
8		cdleme31sc.c	. . 3 ⊢ 𝐶 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊)))
9	8	csbeq2i 3845	. 2 ⊢ ⦋𝑅 / 𝑠⦌𝐶 = ⦋𝑅 / 𝑠⦌((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊)))
10		cdleme31sc.x	. 2 ⊢ 𝑋 = ((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊)))
11	7, 9, 10	3eqtr4g 2796	1 ⊢ (𝑅 ∈ 𝐴 → ⦋𝑅 / 𝑠⦌𝐶 = 𝑋)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ⦋csb 3837  (class class class)co 7367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-iota 6454  df-fv 6506  df-ov 7370

This theorem is referenced by:  cdleme31snd  40832  cdleme31sdnN  40833  cdlemefr44  40871  cdlemefr45e  40874  cdleme48fv  40945  cdleme46fvaw  40947  cdleme48bw  40948  cdleme46fsvlpq  40951  cdlemeg46fvcl  40952  cdlemeg49le  40957  cdlemeg46fjgN  40967  cdlemeg46rjgN  40968  cdlemeg46fjv  40969  cdleme48d  40981  cdlemeg49lebilem  40985  cdleme50eq  40987  cdleme50f  40988  cdlemg2jlemOLDN  41039  cdlemg2klem  41041