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Theorem ifeq2 4530
Description: Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
ifeq2 (𝐴 = 𝐵 → if(𝜑, 𝐶, 𝐴) = if(𝜑, 𝐶, 𝐵))

Proof of Theorem ifeq2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 rabeq 3451 . . 3 (𝐴 = 𝐵 → {𝑥𝐴 ∣ ¬ 𝜑} = {𝑥𝐵 ∣ ¬ 𝜑})
21uneq2d 4168 . 2 (𝐴 = 𝐵 → ({𝑥𝐶𝜑} ∪ {𝑥𝐴 ∣ ¬ 𝜑}) = ({𝑥𝐶𝜑} ∪ {𝑥𝐵 ∣ ¬ 𝜑}))
3 dfif6 4528 . 2 if(𝜑, 𝐶, 𝐴) = ({𝑥𝐶𝜑} ∪ {𝑥𝐴 ∣ ¬ 𝜑})
4 dfif6 4528 . 2 if(𝜑, 𝐶, 𝐵) = ({𝑥𝐶𝜑} ∪ {𝑥𝐵 ∣ ¬ 𝜑})
52, 3, 43eqtr4g 2802 1 (𝐴 = 𝐵 → if(𝜑, 𝐶, 𝐴) = if(𝜑, 𝐶, 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1540  {crab 3436  cun 3949  ifcif 4525
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-un 3956  df-if 4526
This theorem is referenced by:  ifeq12  4544  ifeq2d  4546  ifbieq2i  4551  somincom  6154  mdetunilem9  22626  prmorcht  27221  pclogsum  27259  matunitlindflem1  37623  ftc1anclem6  37705  ftc1anclem8  37707  ftc1anc  37708  hdmap1cbv  41804  reabssgn  43649  hoidmv1le  46609  hoidmvlelem3  46612  vonn0ioo  46702  vonn0icc  46703
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