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Theorem csbif 4110
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 3517 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥if(𝜑, 𝐵, 𝐶) = 𝐴 / 𝑥if(𝜑, 𝐵, 𝐶))
2 dfsbcq2 3420 . . . . 5 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
3 csbeq1 3517 . . . . 5 (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
4 csbeq1 3517 . . . . 5 (𝑦 = 𝐴𝑦 / 𝑥𝐶 = 𝐴 / 𝑥𝐶)
52, 3, 4ifbieq12d 4085 . . . 4 (𝑦 = 𝐴 → if([𝑦 / 𝑥]𝜑, 𝑦 / 𝑥𝐵, 𝑦 / 𝑥𝐶) = if([𝐴 / 𝑥]𝜑, 𝐴 / 𝑥𝐵, 𝐴 / 𝑥𝐶))
61, 5eqeq12d 2636 . . 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 3189 . . . 4 𝑦 ∈ V
8 nfs1v 2436 . . . . 5 𝑥[𝑦 / 𝑥]𝜑
9 nfcsb1v 3530 . . . . 5 𝑥𝑦 / 𝑥𝐵
10 nfcsb1v 3530 . . . . 5 𝑥𝑦 / 𝑥𝐶
118, 9, 10nfif 4087 . . . 4 𝑥if([𝑦 / 𝑥]𝜑, 𝑦 / 𝑥𝐵, 𝑦 / 𝑥𝐶)
12 sbequ12 2108 . . . . 5 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
13 csbeq1a 3523 . . . . 5 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
14 csbeq1a 3523 . . . . 5 (𝑥 = 𝑦𝐶 = 𝑦 / 𝑥𝐶)
1512, 13, 14ifbieq12d 4085 . . . 4 (𝑥 = 𝑦 → if(𝜑, 𝐵, 𝐶) = if([𝑦 / 𝑥]𝜑, 𝑦 / 𝑥𝐵, 𝑦 / 𝑥𝐶))
167, 11, 15csbief 3539 . . 3 𝑦 / 𝑥if(𝜑, 𝐵, 𝐶) = if([𝑦 / 𝑥]𝜑, 𝑦 / 𝑥𝐵, 𝑦 / 𝑥𝐶)
176, 16vtoclg 3252 . 2 (𝐴 ∈ V → 𝐴 / 𝑥if(𝜑, 𝐵, 𝐶) = if([𝐴 / 𝑥]𝜑, 𝐴 / 𝑥𝐵, 𝐴 / 𝑥𝐶))
18 csbprc 3952 . . 3 𝐴 ∈ V → 𝐴 / 𝑥if(𝜑, 𝐵, 𝐶) = ∅)
19 csbprc 3952 . . . . 5 𝐴 ∈ V → 𝐴 / 𝑥𝐵 = ∅)
20 csbprc 3952 . . . . 5 𝐴 ∈ V → 𝐴 / 𝑥𝐶 = ∅)
2119, 20ifeq12d 4078 . . . 4 𝐴 ∈ V → if([𝐴 / 𝑥]𝜑, 𝐴 / 𝑥𝐵, 𝐴 / 𝑥𝐶) = if([𝐴 / 𝑥]𝜑, ∅, ∅))
22 ifid 4097 . . . 4 if([𝐴 / 𝑥]𝜑, ∅, ∅) = ∅
2321, 22syl6req 2672 . . 3 𝐴 ∈ V → ∅ = if([𝐴 / 𝑥]𝜑, 𝐴 / 𝑥𝐵, 𝐴 / 𝑥𝐶))
2418, 23eqtrd 2655 . 2 𝐴 ∈ V → 𝐴 / 𝑥if(𝜑, 𝐵, 𝐶) = if([𝐴 / 𝑥]𝜑, 𝐴 / 𝑥𝐵, 𝐴 / 𝑥𝐶))
2517, 24pm2.61i 176 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 1480  [wsb 1877  wcel 1987  Vcvv 3186  [wsbc 3417  csb 3514  c0 3891  ifcif 4058
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-nul 3892  df-if 4059
This theorem is referenced by:  csbopg  4388  fvmptnn04if  20573  csbrdgg  32807  csbfinxpg  32857  cdlemk40  35685
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