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Theorem cdleme31snd 36461
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 csbnestg 4222 . 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 36459 . . . 4 (𝑆𝐴𝑆 / 𝑣𝑁 = 𝑂)
54csbeq1d 3764 . . 3 (𝑆𝐴𝑆 / 𝑣𝑁 / 𝑡𝐷 = 𝑂 / 𝑡𝐷)
63ovexi 6938 . . . 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 36459 . . . 4 (𝑂 ∈ V → 𝑂 / 𝑡𝐷 = 𝐸)
106, 9ax-mp 5 . . 3 𝑂 / 𝑡𝐷 = 𝐸
115, 10syl6eq 2877 . 2 (𝑆𝐴𝑆 / 𝑣𝑁 / 𝑡𝐷 = 𝐸)
121, 11eqtrd 2861 1 (𝑆𝐴𝑆 / 𝑣𝑁 / 𝑡𝐷 = 𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1658  wcel 2166  Vcvv 3414  csb 3757  (class class class)co 6905
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-nul 5013
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-iota 6086  df-fv 6131  df-ov 6908
This theorem is referenced by:  cdlemeg46ngfr  36593
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