Theorem cdleme31se2 36191

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 2902	. . . . 5 ⊢ Ⅎ𝑡(𝑃 ∨ 𝑄)
2		nfcv 2902	. . . . 5 ⊢ Ⅎ𝑡 ∧
3		nfcsb1v 3690	. . . . . 6 ⊢ Ⅎ𝑡⦋𝑆 / 𝑡⦌𝐷
4		nfcv 2902	. . . . . 6 ⊢ Ⅎ𝑡 ∨
5		nfcv 2902	. . . . . 6 ⊢ Ⅎ𝑡((𝑅 ∨ 𝑆) ∧ 𝑊)
6	3, 4, 5	nfov 6840	. . . . 5 ⊢ Ⅎ𝑡(⦋𝑆 / 𝑡⦌𝐷 ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊))
7	1, 2, 6	nfov 6840	. . . 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 3683	. . . . 5 ⊢ (𝑡 = 𝑆 → 𝐷 = ⦋𝑆 / 𝑡⦌𝐷)
10		oveq2 6822	. . . . . 6 ⊢ (𝑡 = 𝑆 → (𝑅 ∨ 𝑡) = (𝑅 ∨ 𝑆))
11	10	oveq1d 6829	. . . . 5 ⊢ (𝑡 = 𝑆 → ((𝑅 ∨ 𝑡) ∧ 𝑊) = ((𝑅 ∨ 𝑆) ∧ 𝑊))
12	9, 11	oveq12d 6832	. . . 4 ⊢ (𝑡 = 𝑆 → (𝐷 ∨ ((𝑅 ∨ 𝑡) ∧ 𝑊)) = (⦋𝑆 / 𝑡⦌𝐷 ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊)))
13	12	oveq2d 6830	. . 3 ⊢ (𝑡 = 𝑆 → ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑅 ∨ 𝑡) ∧ 𝑊))) = ((𝑃 ∨ 𝑄) ∧ (⦋𝑆 / 𝑡⦌𝐷 ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊))))
14	8, 13	csbiegf 3698	. 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 4136	. 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 2819	1 ⊢ (𝑆 ∈ 𝐴 → ⦋𝑆 / 𝑡⦌𝐸 = 𝑌)

Syntax hints:   → wi 4   = wceq 1632   ∈ wcel 2139  Ⅎwnfc 2889  ⦋csb 3674  (class class class)co 6814

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-iota 6012  df-fv 6057  df-ov 6817

This theorem is referenced by:  cdlemeg47rv2  36318