Theorem cdleme31snd 40762

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 4378	. 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 40760	. . . 4 ⊢ (𝑆 ∈ 𝐴 → ⦋𝑆 / 𝑣⦌𝑁 = 𝑂)
5	4	csbeq1d 3855	. . 3 ⊢ (𝑆 ∈ 𝐴 → ⦋⦋𝑆 / 𝑣⦌𝑁 / 𝑡⦌𝐷 = ⦋𝑂 / 𝑡⦌𝐷)
6	3	ovexi 7402	. . . 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 40760	. . . 4 ⊢ (𝑂 ∈ V → ⦋𝑂 / 𝑡⦌𝐷 = 𝐸)
10	6, 9	ax-mp 5	. . 3 ⊢ ⦋𝑂 / 𝑡⦌𝐷 = 𝐸
11	5, 10	eqtrdi 2788	. 2 ⊢ (𝑆 ∈ 𝐴 → ⦋⦋𝑆 / 𝑣⦌𝑁 / 𝑡⦌𝐷 = 𝐸)
12	1, 11	eqtrd 2772	1 ⊢ (𝑆 ∈ 𝐴 → ⦋𝑆 / 𝑣⦌⦋𝑁 / 𝑡⦌𝐷 = 𝐸)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3442  ⦋csb 3851  (class class class)co 7368

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-nul 5253

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6456  df-fv 6508  df-ov 7371

This theorem is referenced by:  cdlemeg46ngfr  40894