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Theorem cdleme31snd 37627
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 4356 . 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 37625 . . . 4 (𝑆𝐴𝑆 / 𝑣𝑁 = 𝑂)
54csbeq1d 3870 . . 3 (𝑆𝐴𝑆 / 𝑣𝑁 / 𝑡𝐷 = 𝑂 / 𝑡𝐷)
63ovexi 7183 . . . 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 37625 . . . 4 (𝑂 ∈ V → 𝑂 / 𝑡𝐷 = 𝐸)
106, 9ax-mp 5 . . 3 𝑂 / 𝑡𝐷 = 𝐸
115, 10syl6eq 2875 . 2 (𝑆𝐴𝑆 / 𝑣𝑁 / 𝑡𝐷 = 𝐸)
121, 11eqtrd 2859 1 (𝑆𝐴𝑆 / 𝑣𝑁 / 𝑡𝐷 = 𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  Vcvv 3480  csb 3866  (class class class)co 7149
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-nul 5196
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-iota 6302  df-fv 6351  df-ov 7152
This theorem is referenced by:  cdlemeg46ngfr  37759
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