Theorem cdleme31sc 40371

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 2892	. . 3 ⊢ (𝑅 ∈ 𝐴 → Ⅎ𝑠((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊))))
2		oveq1 7376	. . . 4 ⊢ (𝑠 = 𝑅 → (𝑠 ∨ 𝑈) = (𝑅 ∨ 𝑈))
3		oveq2 7377	. . . . . 6 ⊢ (𝑠 = 𝑅 → (𝑃 ∨ 𝑠) = (𝑃 ∨ 𝑅))
4	3	oveq1d 7384	. . . . 5 ⊢ (𝑠 = 𝑅 → ((𝑃 ∨ 𝑠) ∧ 𝑊) = ((𝑃 ∨ 𝑅) ∧ 𝑊))
5	4	oveq2d 7385	. . . 4 ⊢ (𝑠 = 𝑅 → (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊)) = (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊)))
6	2, 5	oveq12d 7387	. . 3 ⊢ (𝑠 = 𝑅 → ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊))) = ((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊))))
7	1, 6	csbiegf 3892	. 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 3867	. 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 2789	1 ⊢ (𝑅 ∈ 𝐴 → ⦋𝑅 / 𝑠⦌𝐶 = 𝑋)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ⦋csb 3859  (class class class)co 7369

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-iota 6452  df-fv 6507  df-ov 7372

This theorem is referenced by:  cdleme31snd  40373  cdleme31sdnN  40374  cdlemefr44  40412  cdlemefr45e  40415  cdleme48fv  40486  cdleme46fvaw  40488  cdleme48bw  40489  cdleme46fsvlpq  40492  cdlemeg46fvcl  40493  cdlemeg49le  40498  cdlemeg46fjgN  40508  cdlemeg46rjgN  40509  cdlemeg46fjv  40510  cdleme48d  40522  cdlemeg49lebilem  40526  cdleme50eq  40528  cdleme50f  40529  cdlemg2jlemOLDN  40580  cdlemg2klem  40582