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Theorem csbif 4088
 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 3502 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥if(𝜑, 𝐵, 𝐶) = 𝐴 / 𝑥if(𝜑, 𝐵, 𝐶))
2 dfsbcq2 3405 . . . . 5 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
3 csbeq1 3502 . . . . 5 (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
4 csbeq1 3502 . . . . 5 (𝑦 = 𝐴𝑦 / 𝑥𝐶 = 𝐴 / 𝑥𝐶)
52, 3, 4ifbieq12d 4063 . . . 4 (𝑦 = 𝐴 → if([𝑦 / 𝑥]𝜑, 𝑦 / 𝑥𝐵, 𝑦 / 𝑥𝐶) = if([𝐴 / 𝑥]𝜑, 𝐴 / 𝑥𝐵, 𝐴 / 𝑥𝐶))
61, 5eqeq12d 2625 . . 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 3176 . . . 4 𝑦 ∈ V
8 nfs1v 2425 . . . . 5 𝑥[𝑦 / 𝑥]𝜑
9 nfcsb1v 3515 . . . . 5 𝑥𝑦 / 𝑥𝐵
10 nfcsb1v 3515 . . . . 5 𝑥𝑦 / 𝑥𝐶
118, 9, 10nfif 4065 . . . 4 𝑥if([𝑦 / 𝑥]𝜑, 𝑦 / 𝑥𝐵, 𝑦 / 𝑥𝐶)
12 sbequ12 2097 . . . . 5 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
13 csbeq1a 3508 . . . . 5 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
14 csbeq1a 3508 . . . . 5 (𝑥 = 𝑦𝐶 = 𝑦 / 𝑥𝐶)
1512, 13, 14ifbieq12d 4063 . . . 4 (𝑥 = 𝑦 → if(𝜑, 𝐵, 𝐶) = if([𝑦 / 𝑥]𝜑, 𝑦 / 𝑥𝐵, 𝑦 / 𝑥𝐶))
167, 11, 15csbief 3524 . . 3 𝑦 / 𝑥if(𝜑, 𝐵, 𝐶) = if([𝑦 / 𝑥]𝜑, 𝑦 / 𝑥𝐵, 𝑦 / 𝑥𝐶)
176, 16vtoclg 3239 . 2 (𝐴 ∈ V → 𝐴 / 𝑥if(𝜑, 𝐵, 𝐶) = if([𝐴 / 𝑥]𝜑, 𝐴 / 𝑥𝐵, 𝐴 / 𝑥𝐶))
18 csbprc 3932 . . 3 𝐴 ∈ V → 𝐴 / 𝑥if(𝜑, 𝐵, 𝐶) = ∅)
19 csbprc 3932 . . . . 5 𝐴 ∈ V → 𝐴 / 𝑥𝐵 = ∅)
20 csbprc 3932 . . . . 5 𝐴 ∈ V → 𝐴 / 𝑥𝐶 = ∅)
2119, 20ifeq12d 4056 . . . 4 𝐴 ∈ V → if([𝐴 / 𝑥]𝜑, 𝐴 / 𝑥𝐵, 𝐴 / 𝑥𝐶) = if([𝐴 / 𝑥]𝜑, ∅, ∅))
22 ifid 4075 . . . 4 if([𝐴 / 𝑥]𝜑, ∅, ∅) = ∅
2321, 22syl6req 2661 . . 3 𝐴 ∈ V → ∅ = if([𝐴 / 𝑥]𝜑, 𝐴 / 𝑥𝐵, 𝐴 / 𝑥𝐶))
2418, 23eqtrd 2644 . 2 𝐴 ∈ V → 𝐴 / 𝑥if(𝜑, 𝐵, 𝐶) = if([𝐴 / 𝑥]𝜑, 𝐴 / 𝑥𝐵, 𝐴 / 𝑥𝐶))
2517, 24pm2.61i 175 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 1475  [wsb 1867   ∈ wcel 1977  Vcvv 3173  [wsbc 3402  ⦋csb 3499  ∅c0 3874  ifcif 4036 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-nul 3875  df-if 4037 This theorem is referenced by:  csbopg  4358  fvmptnn04if  20473  csbrdgg  32351  csbfinxpg  32401  cdlemk40  35223
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