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Theorem cdleme31snd 40368
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 (𝑆𝐴𝑆 / 𝑣𝑁 / 𝑡𝐷 = 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
StepHypRef Expression
1 csbnestgw 4429 . 2 (𝑆𝐴𝑆 / 𝑣𝑁 / 𝑡𝐷 = 𝑆 / 𝑣𝑁 / 𝑡𝐷)
2 cdleme31snd.n . . . . 5 𝑁 = ((𝑣 𝑉) (𝑃 ((𝑄 𝑣) 𝑊)))
3 cdleme31snd.o . . . . 5 𝑂 = ((𝑆 𝑉) (𝑃 ((𝑄 𝑆) 𝑊)))
42, 3cdleme31sc 40366 . . . 4 (𝑆𝐴𝑆 / 𝑣𝑁 = 𝑂)
54csbeq1d 3911 . . 3 (𝑆𝐴𝑆 / 𝑣𝑁 / 𝑡𝐷 = 𝑂 / 𝑡𝐷)
63ovexi 7464 . . . 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)))
97, 8cdleme31sc 40366 . . . 4 (𝑂 ∈ V → 𝑂 / 𝑡𝐷 = 𝐸)
106, 9ax-mp 5 . . 3 𝑂 / 𝑡𝐷 = 𝐸
115, 10eqtrdi 2790 . 2 (𝑆𝐴𝑆 / 𝑣𝑁 / 𝑡𝐷 = 𝐸)
121, 11eqtrd 2774 1 (𝑆𝐴𝑆 / 𝑣𝑁 / 𝑡𝐷 = 𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1536  wcel 2105  Vcvv 3477  csb 3907  (class class class)co 7430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-nul 5311
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-iota 6515  df-fv 6570  df-ov 7433
This theorem is referenced by:  cdlemeg46ngfr  40500
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