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Theorem ifeq1 4229
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 3342 . . 3 (𝐴 = 𝐵 → {𝑥𝐴𝜑} = {𝑥𝐵𝜑})
21uneq1d 3917 . 2 (𝐴 = 𝐵 → ({𝑥𝐴𝜑} ∪ {𝑥𝐶 ∣ ¬ 𝜑}) = ({𝑥𝐵𝜑} ∪ {𝑥𝐶 ∣ ¬ 𝜑}))
3 dfif6 4228 . 2 if(𝜑, 𝐴, 𝐶) = ({𝑥𝐴𝜑} ∪ {𝑥𝐶 ∣ ¬ 𝜑})
4 dfif6 4228 . 2 if(𝜑, 𝐵, 𝐶) = ({𝑥𝐵𝜑} ∪ {𝑥𝐶 ∣ ¬ 𝜑})
52, 3, 43eqtr4g 2830 1 (𝐴 = 𝐵 → if(𝜑, 𝐴, 𝐶) = if(𝜑, 𝐵, 𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1631  {crab 3065  cun 3721  ifcif 4225
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rab 3070  df-v 3353  df-un 3728  df-if 4226
This theorem is referenced by:  ifeq12  4242  ifeq1d  4243  ifbieq12i  4251  ifexg  4296  rdgeq2  7661  dfoi  8572  wemaplem2  8608  cantnflem1  8750  prodeq2w  14849  prodeq2ii  14850  mgm2nsgrplem2  17614  mgm2nsgrplem3  17615  mplcoe3  19681  marrepval0  20585  ellimc  23857  ply1nzb  24102  dchrvmasumiflem1  25411  signspval  30969  dfrdg2  32037  dfafv2  41732
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