Theorem cdleme31se2 38397

Description: Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 3-Apr-2013.)

Hypotheses

Ref	Expression
cdleme31se2.e	⊢ 𝐸 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑅 ∨ 𝑡) ∧ 𝑊)))
cdleme31se2.y	⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∧ (⦋𝑆 / 𝑡⦌𝐷 ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊)))

Assertion

Ref	Expression
cdleme31se2	⊢ (𝑆 ∈ 𝐴 → ⦋𝑆 / 𝑡⦌𝐸 = 𝑌)

Distinct variable groups:   𝑡,𝐴   𝑡, ∨   𝑡, ∧   𝑡,𝑃   𝑡,𝑄   𝑡,𝑅   𝑡,𝑆   𝑡,𝑊

Allowed substitution hints:   𝐷(𝑡)   𝐸(𝑡)   𝑌(𝑡)

Proof of Theorem cdleme31se2

Step	Hyp	Ref	Expression
1		nfcv 2907	. . . . 5 ⊢ Ⅎ𝑡(𝑃 ∨ 𝑄)
2		nfcv 2907	. . . . 5 ⊢ Ⅎ𝑡 ∧
3		nfcsb1v 3857	. . . . . 6 ⊢ Ⅎ𝑡⦋𝑆 / 𝑡⦌𝐷
4		nfcv 2907	. . . . . 6 ⊢ Ⅎ𝑡 ∨
5		nfcv 2907	. . . . . 6 ⊢ Ⅎ𝑡((𝑅 ∨ 𝑆) ∧ 𝑊)
6	3, 4, 5	nfov 7305	. . . . 5 ⊢ Ⅎ𝑡(⦋𝑆 / 𝑡⦌𝐷 ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊))
7	1, 2, 6	nfov 7305	. . . 4 ⊢ Ⅎ𝑡((𝑃 ∨ 𝑄) ∧ (⦋𝑆 / 𝑡⦌𝐷 ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊)))
8	7	a1i 11	. . 3 ⊢ (𝑆 ∈ 𝐴 → Ⅎ𝑡((𝑃 ∨ 𝑄) ∧ (⦋𝑆 / 𝑡⦌𝐷 ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊))))
9		csbeq1a 3846	. . . . 5 ⊢ (𝑡 = 𝑆 → 𝐷 = ⦋𝑆 / 𝑡⦌𝐷)
10		oveq2 7283	. . . . . 6 ⊢ (𝑡 = 𝑆 → (𝑅 ∨ 𝑡) = (𝑅 ∨ 𝑆))
11	10	oveq1d 7290	. . . . 5 ⊢ (𝑡 = 𝑆 → ((𝑅 ∨ 𝑡) ∧ 𝑊) = ((𝑅 ∨ 𝑆) ∧ 𝑊))
12	9, 11	oveq12d 7293	. . . 4 ⊢ (𝑡 = 𝑆 → (𝐷 ∨ ((𝑅 ∨ 𝑡) ∧ 𝑊)) = (⦋𝑆 / 𝑡⦌𝐷 ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊)))
13	12	oveq2d 7291	. . 3 ⊢ (𝑡 = 𝑆 → ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑅 ∨ 𝑡) ∧ 𝑊))) = ((𝑃 ∨ 𝑄) ∧ (⦋𝑆 / 𝑡⦌𝐷 ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊))))
14	8, 13	csbiegf 3866	. 2 ⊢ (𝑆 ∈ 𝐴 → ⦋𝑆 / 𝑡⦌((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑅 ∨ 𝑡) ∧ 𝑊))) = ((𝑃 ∨ 𝑄) ∧ (⦋𝑆 / 𝑡⦌𝐷 ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊))))
15		cdleme31se2.e	. . 3 ⊢ 𝐸 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑅 ∨ 𝑡) ∧ 𝑊)))
16	15	csbeq2i 3840	. 2 ⊢ ⦋𝑆 / 𝑡⦌𝐸 = ⦋𝑆 / 𝑡⦌((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑅 ∨ 𝑡) ∧ 𝑊)))
17		cdleme31se2.y	. 2 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∧ (⦋𝑆 / 𝑡⦌𝐷 ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊)))
18	14, 16, 17	3eqtr4g 2803	1 ⊢ (𝑆 ∈ 𝐴 → ⦋𝑆 / 𝑡⦌𝐸 = 𝑌)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  Ⅎwnfc 2887  ⦋csb 3832  (class class class)co 7275

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-iota 6391  df-fv 6441  df-ov 7278

This theorem is referenced by:  cdlemeg47rv2  38524