Mathbox for Norm Megill < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdleme31se Structured version   Visualization version   GIF version

Theorem cdleme31se 36456
 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 2970 . . 3 (𝑅𝐴𝑠((𝑃 𝑄) (𝐷 ((𝑅 𝑇) 𝑊))))
2 oveq1 6917 . . . . . 6 (𝑠 = 𝑅 → (𝑠 𝑇) = (𝑅 𝑇))
32oveq1d 6925 . . . . 5 (𝑠 = 𝑅 → ((𝑠 𝑇) 𝑊) = ((𝑅 𝑇) 𝑊))
43oveq2d 6926 . . . 4 (𝑠 = 𝑅 → (𝐷 ((𝑠 𝑇) 𝑊)) = (𝐷 ((𝑅 𝑇) 𝑊)))
54oveq2d 6926 . . 3 (𝑠 = 𝑅 → ((𝑃 𝑄) (𝐷 ((𝑠 𝑇) 𝑊))) = ((𝑃 𝑄) (𝐷 ((𝑅 𝑇) 𝑊))))
61, 5csbiegf 3781 . 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 4219 . 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 2886 1 (𝑅𝐴𝑅 / 𝑠𝐸 = 𝑌)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1656   ∈ wcel 2164  ⦋csb 3757  (class class class)co 6910 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-iota 6090  df-fv 6135  df-ov 6913 This theorem is referenced by:  cdleme31sde  36459  cdleme31sn1c  36462
 Copyright terms: Public domain W3C validator