Theorem cdleme31sc 41441

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 2924	. . 3 ⊢ (𝑅 ∈ 𝐴 → Ⅎ𝑠((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊))))
2		oveq1 7427	. . . 4 ⊢ (𝑠 = 𝑅 → (𝑠 ∨ 𝑈) = (𝑅 ∨ 𝑈))
3		oveq2 7428	. . . . . 6 ⊢ (𝑠 = 𝑅 → (𝑃 ∨ 𝑠) = (𝑃 ∨ 𝑅))
4	3	oveq1d 7435	. . . . 5 ⊢ (𝑠 = 𝑅 → ((𝑃 ∨ 𝑠) ∧ 𝑊) = ((𝑃 ∨ 𝑅) ∧ 𝑊))
5	4	oveq2d 7436	. . . 4 ⊢ (𝑠 = 𝑅 → (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊)) = (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊)))
6	2, 5	oveq12d 7438	. . 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 2821	1 ⊢ (𝑅 ∈ 𝐴 → ⦋𝑅 / 𝑠⦌𝐶 = 𝑋)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ⦋csb 3847  (class class class)co 7420

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6494  df-fv 6546  df-ov 7423

This theorem is used by:  cdleme31snd  41443  cdleme31sdnN  41444  cdlemefr44  41482  cdlemefr45e  41485  cdleme48fv  41556  cdleme46fvaw  41558  cdleme48bw  41559  cdleme46fsvlpq  41562  cdlemeg46fvcl  41563  cdlemeg49le  41568  cdlemeg46fjgN  41578  cdlemeg46rjgN  41579  cdlemeg46fjv  41580  cdleme48d  41592  cdlemeg49lebilem  41596  cdleme50eq  41598  cdleme50f  41599  cdlemg2jlemOLDN  41650  cdlemg2klem  41652