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Theorem cdleme31sc 34489
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 2747 . . 3 (𝑅𝐴𝑠((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊))))
2 oveq1 6530 . . . 4 (𝑠 = 𝑅 → (𝑠 𝑈) = (𝑅 𝑈))
3 oveq2 6531 . . . . . 6 (𝑠 = 𝑅 → (𝑃 𝑠) = (𝑃 𝑅))
43oveq1d 6538 . . . . 5 (𝑠 = 𝑅 → ((𝑃 𝑠) 𝑊) = ((𝑃 𝑅) 𝑊))
54oveq2d 6539 . . . 4 (𝑠 = 𝑅 → (𝑄 ((𝑃 𝑠) 𝑊)) = (𝑄 ((𝑃 𝑅) 𝑊)))
62, 5oveq12d 6541 . . 3 (𝑠 = 𝑅 → ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊))) = ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊))))
71, 6csbiegf 3518 . 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 3940 . 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 2664 1 (𝑅𝐴𝑅 / 𝑠𝐶 = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1975  csb 3494  (class class class)co 6523
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-rex 2897  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-br 4574  df-iota 5750  df-fv 5794  df-ov 6526
This theorem is referenced by:  cdleme31snd  34491  cdleme31sdnN  34492  cdlemefr44  34530  cdlemefr45e  34533  cdleme48fv  34604  cdleme46fvaw  34606  cdleme48bw  34607  cdleme46fsvlpq  34610  cdlemeg46fvcl  34611  cdlemeg49le  34616  cdlemeg46fjgN  34626  cdlemeg46rjgN  34627  cdlemeg46fjv  34628  cdleme48d  34640  cdlemeg49lebilem  34644  cdleme50eq  34646  cdleme50f  34647  cdlemg2jlemOLDN  34698  cdlemg2klem  34700
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