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Theorem csbif 4581
Description: Distribute proper substitution through the conditional operator. (Contributed by NM, 24-Feb-2013.) (Revised by NM, 19-Aug-2018.)
Assertion
Ref Expression
csbif 𝐴 / 𝑥if(𝜑, 𝐵, 𝐶) = if([𝐴 / 𝑥]𝜑, 𝐴 / 𝑥𝐵, 𝐴 / 𝑥𝐶)

Proof of Theorem csbif
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3892 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥if(𝜑, 𝐵, 𝐶) = 𝐴 / 𝑥if(𝜑, 𝐵, 𝐶))
2 dfsbcq2 3777 . . . . 5 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
3 csbeq1 3892 . . . . 5 (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
4 csbeq1 3892 . . . . 5 (𝑦 = 𝐴𝑦 / 𝑥𝐶 = 𝐴 / 𝑥𝐶)
52, 3, 4ifbieq12d 4552 . . . 4 (𝑦 = 𝐴 → if([𝑦 / 𝑥]𝜑, 𝑦 / 𝑥𝐵, 𝑦 / 𝑥𝐶) = if([𝐴 / 𝑥]𝜑, 𝐴 / 𝑥𝐵, 𝐴 / 𝑥𝐶))
61, 5eqeq12d 2743 . . 3 (𝑦 = 𝐴 → (𝑦 / 𝑥if(𝜑, 𝐵, 𝐶) = if([𝑦 / 𝑥]𝜑, 𝑦 / 𝑥𝐵, 𝑦 / 𝑥𝐶) ↔ 𝐴 / 𝑥if(𝜑, 𝐵, 𝐶) = if([𝐴 / 𝑥]𝜑, 𝐴 / 𝑥𝐵, 𝐴 / 𝑥𝐶)))
7 vex 3473 . . . 4 𝑦 ∈ V
8 nfs1v 2146 . . . . 5 𝑥[𝑦 / 𝑥]𝜑
9 nfcsb1v 3914 . . . . 5 𝑥𝑦 / 𝑥𝐵
10 nfcsb1v 3914 . . . . 5 𝑥𝑦 / 𝑥𝐶
118, 9, 10nfif 4554 . . . 4 𝑥if([𝑦 / 𝑥]𝜑, 𝑦 / 𝑥𝐵, 𝑦 / 𝑥𝐶)
12 sbequ12 2236 . . . . 5 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
13 csbeq1a 3903 . . . . 5 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
14 csbeq1a 3903 . . . . 5 (𝑥 = 𝑦𝐶 = 𝑦 / 𝑥𝐶)
1512, 13, 14ifbieq12d 4552 . . . 4 (𝑥 = 𝑦 → if(𝜑, 𝐵, 𝐶) = if([𝑦 / 𝑥]𝜑, 𝑦 / 𝑥𝐵, 𝑦 / 𝑥𝐶))
167, 11, 15csbief 3924 . . 3 𝑦 / 𝑥if(𝜑, 𝐵, 𝐶) = if([𝑦 / 𝑥]𝜑, 𝑦 / 𝑥𝐵, 𝑦 / 𝑥𝐶)
176, 16vtoclg 3538 . 2 (𝐴 ∈ V → 𝐴 / 𝑥if(𝜑, 𝐵, 𝐶) = if([𝐴 / 𝑥]𝜑, 𝐴 / 𝑥𝐵, 𝐴 / 𝑥𝐶))
18 csbprc 4402 . . 3 𝐴 ∈ V → 𝐴 / 𝑥if(𝜑, 𝐵, 𝐶) = ∅)
19 csbprc 4402 . . . . 5 𝐴 ∈ V → 𝐴 / 𝑥𝐵 = ∅)
20 csbprc 4402 . . . . 5 𝐴 ∈ V → 𝐴 / 𝑥𝐶 = ∅)
2119, 20ifeq12d 4545 . . . 4 𝐴 ∈ V → if([𝐴 / 𝑥]𝜑, 𝐴 / 𝑥𝐵, 𝐴 / 𝑥𝐶) = if([𝐴 / 𝑥]𝜑, ∅, ∅))
22 ifid 4564 . . . 4 if([𝐴 / 𝑥]𝜑, ∅, ∅) = ∅
2321, 22eqtr2di 2784 . . 3 𝐴 ∈ V → ∅ = if([𝐴 / 𝑥]𝜑, 𝐴 / 𝑥𝐵, 𝐴 / 𝑥𝐶))
2418, 23eqtrd 2767 . 2 𝐴 ∈ V → 𝐴 / 𝑥if(𝜑, 𝐵, 𝐶) = if([𝐴 / 𝑥]𝜑, 𝐴 / 𝑥𝐵, 𝐴 / 𝑥𝐶))
2517, 24pm2.61i 182 1 𝐴 / 𝑥if(𝜑, 𝐵, 𝐶) = if([𝐴 / 𝑥]𝜑, 𝐴 / 𝑥𝐵, 𝐴 / 𝑥𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1534  [wsb 2060  wcel 2099  Vcvv 3469  [wsbc 3774  csb 3889  c0 4318  ifcif 4524
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-nul 4319  df-if 4525
This theorem is referenced by:  csbopg  4887  fvmptnn04if  22738  csbrdgg  36744  csbfinxpg  36803  cdlemk40  40327
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