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Theorem cdleme31sc 36458
Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 31-Mar-2013.)
Hypotheses
Ref Expression
cdleme31sc.c 𝐶 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))
cdleme31sc.x 𝑋 = ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊)))
Assertion
Ref Expression
cdleme31sc (𝑅𝐴𝑅 / 𝑠𝐶 = 𝑋)
Distinct variable groups:   𝐴,𝑠   ,𝑠   ,𝑠   𝑃,𝑠   𝑄,𝑠   𝑅,𝑠   𝑈,𝑠   𝑊,𝑠
Allowed substitution hints:   𝐶(𝑠)   𝑋(𝑠)

Proof of Theorem cdleme31sc
StepHypRef Expression
1 nfcvd 2970 . . 3 (𝑅𝐴𝑠((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊))))
2 oveq1 6917 . . . 4 (𝑠 = 𝑅 → (𝑠 𝑈) = (𝑅 𝑈))
3 oveq2 6918 . . . . . 6 (𝑠 = 𝑅 → (𝑃 𝑠) = (𝑃 𝑅))
43oveq1d 6925 . . . . 5 (𝑠 = 𝑅 → ((𝑃 𝑠) 𝑊) = ((𝑃 𝑅) 𝑊))
54oveq2d 6926 . . . 4 (𝑠 = 𝑅 → (𝑄 ((𝑃 𝑠) 𝑊)) = (𝑄 ((𝑃 𝑅) 𝑊)))
62, 5oveq12d 6928 . . 3 (𝑠 = 𝑅 → ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊))) = ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊))))
71, 6csbiegf 3781 . 2 (𝑅𝐴𝑅 / 𝑠((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊))) = ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊))))
8 cdleme31sc.c . . 3 𝐶 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))
98csbeq2i 4219 . 2 𝑅 / 𝑠𝐶 = 𝑅 / 𝑠((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))
10 cdleme31sc.x . 2 𝑋 = ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊)))
117, 9, 103eqtr4g 2886 1 (𝑅𝐴𝑅 / 𝑠𝐶 = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1656  wcel 2164  csb 3757  (class class class)co 6910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  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 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-iota 6090  df-fv 6135  df-ov 6913
This theorem is referenced by:  cdleme31snd  36460  cdleme31sdnN  36461  cdlemefr44  36499  cdlemefr45e  36502  cdleme48fv  36573  cdleme46fvaw  36575  cdleme48bw  36576  cdleme46fsvlpq  36579  cdlemeg46fvcl  36580  cdlemeg49le  36585  cdlemeg46fjgN  36595  cdlemeg46rjgN  36596  cdlemeg46fjv  36597  cdleme48d  36609  cdlemeg49lebilem  36613  cdleme50eq  36615  cdleme50f  36616  cdlemg2jlemOLDN  36667  cdlemg2klem  36669
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