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Theorem ifeq2 3553
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 2744 . . 3 (𝐴 = 𝐵 → {𝑥𝐴 ∣ ¬ 𝜑} = {𝑥𝐵 ∣ ¬ 𝜑})
21uneq2d 3304 . 2 (𝐴 = 𝐵 → ({𝑥𝐶𝜑} ∪ {𝑥𝐴 ∣ ¬ 𝜑}) = ({𝑥𝐶𝜑} ∪ {𝑥𝐵 ∣ ¬ 𝜑}))
3 dfif6 3551 . 2 if(𝜑, 𝐶, 𝐴) = ({𝑥𝐶𝜑} ∪ {𝑥𝐴 ∣ ¬ 𝜑})
4 dfif6 3551 . 2 if(𝜑, 𝐶, 𝐵) = ({𝑥𝐶𝜑} ∪ {𝑥𝐵 ∣ ¬ 𝜑})
52, 3, 43eqtr4g 2247 1 (𝐴 = 𝐵 → if(𝜑, 𝐶, 𝐴) = if(𝜑, 𝐶, 𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1364  {crab 2472  cun 3142  ifcif 3549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rab 2477  df-v 2754  df-un 3148  df-if 3550
This theorem is referenced by:  ifeq12  3565  ifeq2d  3567  ifbieq2i  3572  xrmaxiflemcom  11288
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