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Theorem cdleme31sn 37625
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 1916 . . . . 5 𝑠 𝑅 (𝑃 𝑄)
2 nfcsb1v 3890 . . . . 5 𝑠𝑅 / 𝑠𝐼
3 nfcsb1v 3890 . . . . 5 𝑠𝑅 / 𝑠𝐷
41, 2, 3nfif 4479 . . . 4 𝑠if(𝑅 (𝑃 𝑄), 𝑅 / 𝑠𝐼, 𝑅 / 𝑠𝐷)
54a1i 11 . . 3 (𝑅𝐴𝑠if(𝑅 (𝑃 𝑄), 𝑅 / 𝑠𝐼, 𝑅 / 𝑠𝐷))
6 breq1 5055 . . . 4 (𝑠 = 𝑅 → (𝑠 (𝑃 𝑄) ↔ 𝑅 (𝑃 𝑄)))
7 csbeq1a 3880 . . . 4 (𝑠 = 𝑅𝐼 = 𝑅 / 𝑠𝐼)
8 csbeq1a 3880 . . . 4 (𝑠 = 𝑅𝐷 = 𝑅 / 𝑠𝐷)
96, 7, 8ifbieq12d 4477 . . 3 (𝑠 = 𝑅 → if(𝑠 (𝑃 𝑄), 𝐼, 𝐷) = if(𝑅 (𝑃 𝑄), 𝑅 / 𝑠𝐼, 𝑅 / 𝑠𝐷))
105, 9csbiegf 3899 . 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 3874 . 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 2884 1 (𝑅𝐴𝑅 / 𝑠𝑁 = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  wnfc 2962  csb 3866  ifcif 4450   class class class wbr 5052  (class class class)co 7149
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
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 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-un 3924  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-br 5053
This theorem is referenced by:  cdleme31sn1  37626  cdleme31sn2  37634
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