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Theorem ifeq1 4062
Description: Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
ifeq1 (𝐴 = 𝐵 → if(𝜑, 𝐴, 𝐶) = if(𝜑, 𝐵, 𝐶))

Proof of Theorem ifeq1
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 rabeq 3179 . . 3 (𝐴 = 𝐵 → {𝑥𝐴𝜑} = {𝑥𝐵𝜑})
21uneq1d 3744 . 2 (𝐴 = 𝐵 → ({𝑥𝐴𝜑} ∪ {𝑥𝐶 ∣ ¬ 𝜑}) = ({𝑥𝐵𝜑} ∪ {𝑥𝐶 ∣ ¬ 𝜑}))
3 dfif6 4061 . 2 if(𝜑, 𝐴, 𝐶) = ({𝑥𝐴𝜑} ∪ {𝑥𝐶 ∣ ¬ 𝜑})
4 dfif6 4061 . 2 if(𝜑, 𝐵, 𝐶) = ({𝑥𝐵𝜑} ∪ {𝑥𝐶 ∣ ¬ 𝜑})
52, 3, 43eqtr4g 2680 1 (𝐴 = 𝐵 → if(𝜑, 𝐴, 𝐶) = if(𝜑, 𝐵, 𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1480  {crab 2911  cun 3553  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-tru 1483  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-un 3560  df-if 4059
This theorem is referenced by:  ifeq12  4075  ifeq1d  4076  ifbieq12i  4084  ifexg  4129  rdgeq2  7453  dfoi  8360  wemaplem2  8396  cantnflem1  8530  prodeq2w  14567  prodeq2ii  14568  mgm2nsgrplem2  17327  mgm2nsgrplem3  17328  mplcoe3  19385  marrepval0  20286  ellimc  23543  ply1nzb  23786  dchrvmasumiflem1  25090  signspval  30409  dfrdg2  31402  dfafv2  40516
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