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

Theorem cdleme31se 39766
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 2898 . . 3 (𝑅𝐴𝑠((𝑃 𝑄) (𝐷 ((𝑅 𝑇) 𝑊))))
2 oveq1 7412 . . . . . 6 (𝑠 = 𝑅 → (𝑠 𝑇) = (𝑅 𝑇))
32oveq1d 7420 . . . . 5 (𝑠 = 𝑅 → ((𝑠 𝑇) 𝑊) = ((𝑅 𝑇) 𝑊))
43oveq2d 7421 . . . 4 (𝑠 = 𝑅 → (𝐷 ((𝑠 𝑇) 𝑊)) = (𝐷 ((𝑅 𝑇) 𝑊)))
54oveq2d 7421 . . 3 (𝑠 = 𝑅 → ((𝑃 𝑄) (𝐷 ((𝑠 𝑇) 𝑊))) = ((𝑃 𝑄) (𝐷 ((𝑅 𝑇) 𝑊))))
61, 5csbiegf 3922 . 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 3896 . 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 2791 1 (𝑅𝐴𝑅 / 𝑠𝐸 = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  csb 3888  (class class class)co 7405
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-iota 6489  df-fv 6545  df-ov 7408
This theorem is referenced by:  cdleme31sde  39769  cdleme31sn1c  39772
  Copyright terms: Public domain W3C validator