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Theorem cdleme31sn 39555
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 3918 . . . . 5 𝑠𝑅 / 𝑠𝐼
3 nfcsb1v 3918 . . . . 5 𝑠𝑅 / 𝑠𝐷
41, 2, 3nfif 4558 . . . 4 𝑠if(𝑅 (𝑃 𝑄), 𝑅 / 𝑠𝐼, 𝑅 / 𝑠𝐷)
54a1i 11 . . 3 (𝑅𝐴𝑠if(𝑅 (𝑃 𝑄), 𝑅 / 𝑠𝐼, 𝑅 / 𝑠𝐷))
6 breq1 5151 . . . 4 (𝑠 = 𝑅 → (𝑠 (𝑃 𝑄) ↔ 𝑅 (𝑃 𝑄)))
7 csbeq1a 3907 . . . 4 (𝑠 = 𝑅𝐼 = 𝑅 / 𝑠𝐼)
8 csbeq1a 3907 . . . 4 (𝑠 = 𝑅𝐷 = 𝑅 / 𝑠𝐷)
96, 7, 8ifbieq12d 4556 . . 3 (𝑠 = 𝑅 → if(𝑠 (𝑃 𝑄), 𝐼, 𝐷) = if(𝑅 (𝑃 𝑄), 𝑅 / 𝑠𝐼, 𝑅 / 𝑠𝐷))
105, 9csbiegf 3927 . 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 3901 . 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 2796 1 (𝑅𝐴𝑅 / 𝑠𝑁 = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2105  wnfc 2882  csb 3893  ifcif 4528   class class class wbr 5148  (class class class)co 7412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149
This theorem is referenced by:  cdleme31sn1  39556  cdleme31sn2  39564
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