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Theorem cdleme31sc 35491
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 2763 . . 3 (𝑅𝐴𝑠((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊))))
2 oveq1 6642 . . . 4 (𝑠 = 𝑅 → (𝑠 𝑈) = (𝑅 𝑈))
3 oveq2 6643 . . . . . 6 (𝑠 = 𝑅 → (𝑃 𝑠) = (𝑃 𝑅))
43oveq1d 6650 . . . . 5 (𝑠 = 𝑅 → ((𝑃 𝑠) 𝑊) = ((𝑃 𝑅) 𝑊))
54oveq2d 6651 . . . 4 (𝑠 = 𝑅 → (𝑄 ((𝑃 𝑠) 𝑊)) = (𝑄 ((𝑃 𝑅) 𝑊)))
62, 5oveq12d 6653 . . 3 (𝑠 = 𝑅 → ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊))) = ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊))))
71, 6csbiegf 3550 . 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 3984 . 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 2679 1 (𝑅𝐴𝑅 / 𝑠𝐶 = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1481  wcel 1988  csb 3526  (class class class)co 6635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-iota 5839  df-fv 5884  df-ov 6638
This theorem is referenced by:  cdleme31snd  35493  cdleme31sdnN  35494  cdlemefr44  35532  cdlemefr45e  35535  cdleme48fv  35606  cdleme46fvaw  35608  cdleme48bw  35609  cdleme46fsvlpq  35612  cdlemeg46fvcl  35613  cdlemeg49le  35618  cdlemeg46fjgN  35628  cdlemeg46rjgN  35629  cdlemeg46fjv  35630  cdleme48d  35642  cdlemeg49lebilem  35646  cdleme50eq  35648  cdleme50f  35649  cdlemg2jlemOLDN  35700  cdlemg2klem  35702
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