Theorem cdleme31sde 41159

Description: Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 31-Mar-2013.)

Hypotheses

Ref	Expression
cdleme31sde.c	⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
cdleme31sde.e	⊢ 𝐸 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊)))
cdleme31sde.x	⊢ 𝑌 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊)))
cdleme31sde.z	⊢ 𝑍 = ((𝑃 ∨ 𝑄) ∧ (𝑌 ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊)))

Assertion

Ref	Expression
cdleme31sde	⊢ ((𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → ⦋𝑅 / 𝑠⦌⦋𝑆 / 𝑡⦌𝐸 = 𝑍)

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

Allowed substitution hints:   𝐷(𝑡,𝑠)   𝑅(𝑡)   𝑈(𝑡,𝑠)   𝐸(𝑡,𝑠)   𝑍(𝑡,𝑠)

Proof of Theorem cdleme31sde

Step	Hyp	Ref	Expression
1		cdleme31sde.e	. . . . 5 ⊢ 𝐸 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊)))
2	1	csbeq2i 3861	. . . 4 ⊢ ⦋𝑆 / 𝑡⦌𝐸 = ⦋𝑆 / 𝑡⦌((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊)))
3		nfcvd 2926	. . . . 5 ⊢ (𝑆 ∈ 𝐴 → Ⅎ𝑡((𝑃 ∨ 𝑄) ∧ (𝑌 ∨ ((𝑠 ∨ 𝑆) ∧ 𝑊))))
4		oveq1 7417	. . . . . . . . 9 ⊢ (𝑡 = 𝑆 → (𝑡 ∨ 𝑈) = (𝑆 ∨ 𝑈))
5		oveq2 7418	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑆 → (𝑃 ∨ 𝑡) = (𝑃 ∨ 𝑆))
6	5	oveq1d 7425	. . . . . . . . . 10 ⊢ (𝑡 = 𝑆 → ((𝑃 ∨ 𝑡) ∧ 𝑊) = ((𝑃 ∨ 𝑆) ∧ 𝑊))
7	6	oveq2d 7426	. . . . . . . . 9 ⊢ (𝑡 = 𝑆 → (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)) = (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊)))
8	4, 7	oveq12d 7428	. . . . . . . 8 ⊢ (𝑡 = 𝑆 → ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))) = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))))
9		cdleme31sde.c	. . . . . . . 8 ⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
10		cdleme31sde.x	. . . . . . . 8 ⊢ 𝑌 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊)))
11	8, 9, 10	3eqtr4g 2823	. . . . . . 7 ⊢ (𝑡 = 𝑆 → 𝐷 = 𝑌)
12		oveq2 7418	. . . . . . . 8 ⊢ (𝑡 = 𝑆 → (𝑠 ∨ 𝑡) = (𝑠 ∨ 𝑆))
13	12	oveq1d 7425	. . . . . . 7 ⊢ (𝑡 = 𝑆 → ((𝑠 ∨ 𝑡) ∧ 𝑊) = ((𝑠 ∨ 𝑆) ∧ 𝑊))
14	11, 13	oveq12d 7428	. . . . . 6 ⊢ (𝑡 = 𝑆 → (𝐷 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊)) = (𝑌 ∨ ((𝑠 ∨ 𝑆) ∧ 𝑊)))
15	14	oveq2d 7426	. . . . 5 ⊢ (𝑡 = 𝑆 → ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊))) = ((𝑃 ∨ 𝑄) ∧ (𝑌 ∨ ((𝑠 ∨ 𝑆) ∧ 𝑊))))
16	3, 15	csbiegf 3886	. . . 4 ⊢ (𝑆 ∈ 𝐴 → ⦋𝑆 / 𝑡⦌((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊))) = ((𝑃 ∨ 𝑄) ∧ (𝑌 ∨ ((𝑠 ∨ 𝑆) ∧ 𝑊))))
17	2, 16	eqtrid 2810	. . 3 ⊢ (𝑆 ∈ 𝐴 → ⦋𝑆 / 𝑡⦌𝐸 = ((𝑃 ∨ 𝑄) ∧ (𝑌 ∨ ((𝑠 ∨ 𝑆) ∧ 𝑊))))
18	17	csbeq2dv 3860	. 2 ⊢ (𝑆 ∈ 𝐴 → ⦋𝑅 / 𝑠⦌⦋𝑆 / 𝑡⦌𝐸 = ⦋𝑅 / 𝑠⦌((𝑃 ∨ 𝑄) ∧ (𝑌 ∨ ((𝑠 ∨ 𝑆) ∧ 𝑊))))
19		eqid 2763	. . 3 ⊢ ((𝑃 ∨ 𝑄) ∧ (𝑌 ∨ ((𝑠 ∨ 𝑆) ∧ 𝑊))) = ((𝑃 ∨ 𝑄) ∧ (𝑌 ∨ ((𝑠 ∨ 𝑆) ∧ 𝑊)))
20		cdleme31sde.z	. . 3 ⊢ 𝑍 = ((𝑃 ∨ 𝑄) ∧ (𝑌 ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊)))
21	19, 20	cdleme31se 41156	. 2 ⊢ (𝑅 ∈ 𝐴 → ⦋𝑅 / 𝑠⦌((𝑃 ∨ 𝑄) ∧ (𝑌 ∨ ((𝑠 ∨ 𝑆) ∧ 𝑊))) = 𝑍)
22	18, 21	sylan9eqr 2820	1 ⊢ ((𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → ⦋𝑅 / 𝑠⦌⦋𝑆 / 𝑡⦌𝐸 = 𝑍)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ⦋csb 3853  (class class class)co 7410

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-iota 6492  df-fv 6544  df-ov 7413

This theorem is referenced by:  cdlemefs44  41200  cdlemefs45ee  41204  cdleme17d2  41269