Theorem cdleme31sc 40583

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 2897	. . 3 ⊢ (𝑅 ∈ 𝐴 → Ⅎ𝑠((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊))))
2		oveq1 7363	. . . 4 ⊢ (𝑠 = 𝑅 → (𝑠 ∨ 𝑈) = (𝑅 ∨ 𝑈))
3		oveq2 7364	. . . . . 6 ⊢ (𝑠 = 𝑅 → (𝑃 ∨ 𝑠) = (𝑃 ∨ 𝑅))
4	3	oveq1d 7371	. . . . 5 ⊢ (𝑠 = 𝑅 → ((𝑃 ∨ 𝑠) ∧ 𝑊) = ((𝑃 ∨ 𝑅) ∧ 𝑊))
5	4	oveq2d 7372	. . . 4 ⊢ (𝑠 = 𝑅 → (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊)) = (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊)))
6	2, 5	oveq12d 7374	. . 3 ⊢ (𝑠 = 𝑅 → ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊))) = ((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊))))
7	1, 6	csbiegf 3880	. 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 3855	. 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 2794	1 ⊢ (𝑅 ∈ 𝐴 → ⦋𝑅 / 𝑠⦌𝐶 = 𝑋)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  ⦋csb 3847  (class class class)co 7356

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-iota 6446  df-fv 6498  df-ov 7359

This theorem is referenced by:  cdleme31snd  40585  cdleme31sdnN  40586  cdlemefr44  40624  cdlemefr45e  40627  cdleme48fv  40698  cdleme46fvaw  40700  cdleme48bw  40701  cdleme46fsvlpq  40704  cdlemeg46fvcl  40705  cdlemeg49le  40710  cdlemeg46fjgN  40720  cdlemeg46rjgN  40721  cdlemeg46fjv  40722  cdleme48d  40734  cdlemeg49lebilem  40738  cdleme50eq  40740  cdleme50f  40741  cdlemg2jlemOLDN  40792  cdlemg2klem  40794