Theorem cdleme31snd 38400

Description: Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 1-Apr-2013.)

Hypotheses

Ref	Expression
cdleme31snd.d	⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
cdleme31snd.n	⊢ 𝑁 = ((𝑣 ∨ 𝑉) ∧ (𝑃 ∨ ((𝑄 ∨ 𝑣) ∧ 𝑊)))
cdleme31snd.e	⊢ 𝐸 = ((𝑂 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑂) ∧ 𝑊)))
cdleme31snd.o	⊢ 𝑂 = ((𝑆 ∨ 𝑉) ∧ (𝑃 ∨ ((𝑄 ∨ 𝑆) ∧ 𝑊)))

Assertion

Ref	Expression
cdleme31snd	⊢ (𝑆 ∈ 𝐴 → ⦋𝑆 / 𝑣⦌⦋𝑁 / 𝑡⦌𝐷 = 𝐸)

Distinct variable groups:   𝑣,𝐴   𝑣,𝐷   𝑣,𝑡, ∨   𝑡, ∧ ,𝑣   𝑡,𝑂   𝑡,𝑃,𝑣   𝑡,𝑄,𝑣   𝑣,𝑆   𝑡,𝑈,𝑣   𝑣,𝑉   𝑡,𝑊,𝑣

Allowed substitution hints:   𝐴(𝑡)   𝐷(𝑡)   𝑆(𝑡)   𝐸(𝑣,𝑡)   𝑁(𝑣,𝑡)   𝑂(𝑣)   𝑉(𝑡)

Proof of Theorem cdleme31snd

Step	Hyp	Ref	Expression
1		csbnestgw 4355	. 2 ⊢ (𝑆 ∈ 𝐴 → ⦋𝑆 / 𝑣⦌⦋𝑁 / 𝑡⦌𝐷 = ⦋⦋𝑆 / 𝑣⦌𝑁 / 𝑡⦌𝐷)
2		cdleme31snd.n	. . . . 5 ⊢ 𝑁 = ((𝑣 ∨ 𝑉) ∧ (𝑃 ∨ ((𝑄 ∨ 𝑣) ∧ 𝑊)))
3		cdleme31snd.o	. . . . 5 ⊢ 𝑂 = ((𝑆 ∨ 𝑉) ∧ (𝑃 ∨ ((𝑄 ∨ 𝑆) ∧ 𝑊)))
4	2, 3	cdleme31sc 38398	. . . 4 ⊢ (𝑆 ∈ 𝐴 → ⦋𝑆 / 𝑣⦌𝑁 = 𝑂)
5	4	csbeq1d 3836	. . 3 ⊢ (𝑆 ∈ 𝐴 → ⦋⦋𝑆 / 𝑣⦌𝑁 / 𝑡⦌𝐷 = ⦋𝑂 / 𝑡⦌𝐷)
6	3	ovexi 7309	. . . 4 ⊢ 𝑂 ∈ V
7		cdleme31snd.d	. . . . 5 ⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
8		cdleme31snd.e	. . . . 5 ⊢ 𝐸 = ((𝑂 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑂) ∧ 𝑊)))
9	7, 8	cdleme31sc 38398	. . . 4 ⊢ (𝑂 ∈ V → ⦋𝑂 / 𝑡⦌𝐷 = 𝐸)
10	6, 9	ax-mp 5	. . 3 ⊢ ⦋𝑂 / 𝑡⦌𝐷 = 𝐸
11	5, 10	eqtrdi 2794	. 2 ⊢ (𝑆 ∈ 𝐴 → ⦋⦋𝑆 / 𝑣⦌𝑁 / 𝑡⦌𝐷 = 𝐸)
12	1, 11	eqtrd 2778	1 ⊢ (𝑆 ∈ 𝐴 → ⦋𝑆 / 𝑣⦌⦋𝑁 / 𝑡⦌𝐷 = 𝐸)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  Vcvv 3432  ⦋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  ax-nul 5230

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-mo 2540  df-eu 2569  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:  cdlemeg46ngfr  38532