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Theorem cdleme31se 38874
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 2909 . . 3 (𝑅𝐴𝑠((𝑃 𝑄) (𝐷 ((𝑅 𝑇) 𝑊))))
2 oveq1 7369 . . . . . 6 (𝑠 = 𝑅 → (𝑠 𝑇) = (𝑅 𝑇))
32oveq1d 7377 . . . . 5 (𝑠 = 𝑅 → ((𝑠 𝑇) 𝑊) = ((𝑅 𝑇) 𝑊))
43oveq2d 7378 . . . 4 (𝑠 = 𝑅 → (𝐷 ((𝑠 𝑇) 𝑊)) = (𝐷 ((𝑅 𝑇) 𝑊)))
54oveq2d 7378 . . 3 (𝑠 = 𝑅 → ((𝑃 𝑄) (𝐷 ((𝑠 𝑇) 𝑊))) = ((𝑃 𝑄) (𝐷 ((𝑅 𝑇) 𝑊))))
61, 5csbiegf 3894 . 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 3868 . 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 2802 1 (𝑅𝐴𝑅 / 𝑠𝐸 = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  csb 3860  (class class class)co 7362
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-iota 6453  df-fv 6509  df-ov 7365
This theorem is referenced by:  cdleme31sde  38877  cdleme31sn1c  38880
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