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Theorem cdleme31sc 36174
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 2903 . . 3 (𝑅𝐴𝑠((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊))))
2 oveq1 6820 . . . 4 (𝑠 = 𝑅 → (𝑠 𝑈) = (𝑅 𝑈))
3 oveq2 6821 . . . . . 6 (𝑠 = 𝑅 → (𝑃 𝑠) = (𝑃 𝑅))
43oveq1d 6828 . . . . 5 (𝑠 = 𝑅 → ((𝑃 𝑠) 𝑊) = ((𝑃 𝑅) 𝑊))
54oveq2d 6829 . . . 4 (𝑠 = 𝑅 → (𝑄 ((𝑃 𝑠) 𝑊)) = (𝑄 ((𝑃 𝑅) 𝑊)))
62, 5oveq12d 6831 . . 3 (𝑠 = 𝑅 → ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊))) = ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊))))
71, 6csbiegf 3698 . 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 4136 . 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 2819 1 (𝑅𝐴𝑅 / 𝑠𝐶 = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  csb 3674  (class class class)co 6813
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-iota 6012  df-fv 6057  df-ov 6816
This theorem is referenced by:  cdleme31snd  36176  cdleme31sdnN  36177  cdlemefr44  36215  cdlemefr45e  36218  cdleme48fv  36289  cdleme46fvaw  36291  cdleme48bw  36292  cdleme46fsvlpq  36295  cdlemeg46fvcl  36296  cdlemeg49le  36301  cdlemeg46fjgN  36311  cdlemeg46rjgN  36312  cdlemeg46fjv  36313  cdleme48d  36325  cdlemeg49lebilem  36329  cdleme50eq  36331  cdleme50f  36332  cdlemg2jlemOLDN  36383  cdlemg2klem  36385
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