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Theorem cdleme31snd 34692
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 3950 . 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 34690 . . . 4 (𝑆𝐴𝑆 / 𝑣𝑁 = 𝑂)
54csbeq1d 3506 . . 3 (𝑆𝐴𝑆 / 𝑣𝑁 / 𝑡𝐷 = 𝑂 / 𝑡𝐷)
6 ovex 6577 . . . . 5 ((𝑆 𝑉) (𝑃 ((𝑄 𝑆) 𝑊))) ∈ V
73, 6eqeltri 2684 . . . 4 𝑂 ∈ V
8 cdleme31snd.d . . . . 5 𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
9 cdleme31snd.e . . . . 5 𝐸 = ((𝑂 𝑈) (𝑄 ((𝑃 𝑂) 𝑊)))
108, 9cdleme31sc 34690 . . . 4 (𝑂 ∈ V → 𝑂 / 𝑡𝐷 = 𝐸)
117, 10ax-mp 5 . . 3 𝑂 / 𝑡𝐷 = 𝐸
125, 11syl6eq 2660 . 2 (𝑆𝐴𝑆 / 𝑣𝑁 / 𝑡𝐷 = 𝐸)
131, 12eqtrd 2644 1 (𝑆𝐴𝑆 / 𝑣𝑁 / 𝑡𝐷 = 𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1475  wcel 1977  Vcvv 3173  csb 3499  (class class class)co 6549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552
This theorem is referenced by:  cdlemeg46ngfr  34824
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