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Theorem cdleme31sn 34482
Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 26-Feb-2013.)
Hypotheses
Ref Expression
cdleme31sn.n 𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐷)
cdleme31sn.c 𝐶 = if(𝑅 (𝑃 𝑄), 𝑅 / 𝑠𝐼, 𝑅 / 𝑠𝐷)
Assertion
Ref Expression
cdleme31sn (𝑅𝐴𝑅 / 𝑠𝑁 = 𝐶)
Distinct variable groups:   𝐴,𝑠   ,𝑠   ,𝑠   𝑃,𝑠   𝑄,𝑠   𝑅,𝑠
Allowed substitution hints:   𝐶(𝑠)   𝐷(𝑠)   𝐼(𝑠)   𝑁(𝑠)

Proof of Theorem cdleme31sn
StepHypRef Expression
1 nfv 1829 . . . . 5 𝑠 𝑅 (𝑃 𝑄)
2 nfcsb1v 3514 . . . . 5 𝑠𝑅 / 𝑠𝐼
3 nfcsb1v 3514 . . . . 5 𝑠𝑅 / 𝑠𝐷
41, 2, 3nfif 4064 . . . 4 𝑠if(𝑅 (𝑃 𝑄), 𝑅 / 𝑠𝐼, 𝑅 / 𝑠𝐷)
54a1i 11 . . 3 (𝑅𝐴𝑠if(𝑅 (𝑃 𝑄), 𝑅 / 𝑠𝐼, 𝑅 / 𝑠𝐷))
6 breq1 4580 . . . 4 (𝑠 = 𝑅 → (𝑠 (𝑃 𝑄) ↔ 𝑅 (𝑃 𝑄)))
7 csbeq1a 3507 . . . 4 (𝑠 = 𝑅𝐼 = 𝑅 / 𝑠𝐼)
8 csbeq1a 3507 . . . 4 (𝑠 = 𝑅𝐷 = 𝑅 / 𝑠𝐷)
96, 7, 8ifbieq12d 4062 . . 3 (𝑠 = 𝑅 → if(𝑠 (𝑃 𝑄), 𝐼, 𝐷) = if(𝑅 (𝑃 𝑄), 𝑅 / 𝑠𝐼, 𝑅 / 𝑠𝐷))
105, 9csbiegf 3522 . 2 (𝑅𝐴𝑅 / 𝑠if(𝑠 (𝑃 𝑄), 𝐼, 𝐷) = if(𝑅 (𝑃 𝑄), 𝑅 / 𝑠𝐼, 𝑅 / 𝑠𝐷))
11 cdleme31sn.n . . 3 𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐷)
1211csbeq2i 3944 . 2 𝑅 / 𝑠𝑁 = 𝑅 / 𝑠if(𝑠 (𝑃 𝑄), 𝐼, 𝐷)
13 cdleme31sn.c . 2 𝐶 = if(𝑅 (𝑃 𝑄), 𝑅 / 𝑠𝐼, 𝑅 / 𝑠𝐷)
1410, 12, 133eqtr4g 2668 1 (𝑅𝐴𝑅 / 𝑠𝑁 = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1976  wnfc 2737  csb 3498  ifcif 4035   class class class wbr 4577  (class class class)co 6527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578
This theorem is referenced by:  cdleme31sn1  34483  cdleme31sn2  34491
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