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Theorem cdleme31se 37636
Description: Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 26-Feb-2013.)
Hypotheses
Ref Expression
cdleme31se.e 𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑇) 𝑊)))
cdleme31se.y 𝑌 = ((𝑃 𝑄) (𝐷 ((𝑅 𝑇) 𝑊)))
Assertion
Ref Expression
cdleme31se (𝑅𝐴𝑅 / 𝑠𝐸 = 𝑌)
Distinct variable groups:   𝐴,𝑠   𝐷,𝑠   ,𝑠   ,𝑠   𝑃,𝑠   𝑄,𝑠   𝑅,𝑠   𝑊,𝑠   𝑇,𝑠
Allowed substitution hints:   𝐸(𝑠)   𝑌(𝑠)

Proof of Theorem cdleme31se
StepHypRef Expression
1 nfcvd 2980 . . 3 (𝑅𝐴𝑠((𝑃 𝑄) (𝐷 ((𝑅 𝑇) 𝑊))))
2 oveq1 7147 . . . . . 6 (𝑠 = 𝑅 → (𝑠 𝑇) = (𝑅 𝑇))
32oveq1d 7155 . . . . 5 (𝑠 = 𝑅 → ((𝑠 𝑇) 𝑊) = ((𝑅 𝑇) 𝑊))
43oveq2d 7156 . . . 4 (𝑠 = 𝑅 → (𝐷 ((𝑠 𝑇) 𝑊)) = (𝐷 ((𝑅 𝑇) 𝑊)))
54oveq2d 7156 . . 3 (𝑠 = 𝑅 → ((𝑃 𝑄) (𝐷 ((𝑠 𝑇) 𝑊))) = ((𝑃 𝑄) (𝐷 ((𝑅 𝑇) 𝑊))))
61, 5csbiegf 3888 . 2 (𝑅𝐴𝑅 / 𝑠((𝑃 𝑄) (𝐷 ((𝑠 𝑇) 𝑊))) = ((𝑃 𝑄) (𝐷 ((𝑅 𝑇) 𝑊))))
7 cdleme31se.e . . 3 𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑇) 𝑊)))
87csbeq2i 3863 . 2 𝑅 / 𝑠𝐸 = 𝑅 / 𝑠((𝑃 𝑄) (𝐷 ((𝑠 𝑇) 𝑊)))
9 cdleme31se.y . 2 𝑌 = ((𝑃 𝑄) (𝐷 ((𝑅 𝑇) 𝑊)))
106, 8, 93eqtr4g 2882 1 (𝑅𝐴𝑅 / 𝑠𝐸 = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2114  csb 3855  (class class class)co 7140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-v 3471  df-sbc 3748  df-csb 3856  df-un 3913  df-in 3915  df-ss 3925  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-iota 6293  df-fv 6342  df-ov 7143
This theorem is referenced by:  cdleme31sde  37639  cdleme31sn1c  37642
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