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Theorem ifeq2 3417
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 2625 . . 3 (𝐴 = 𝐵 → {𝑥𝐴 ∣ ¬ 𝜑} = {𝑥𝐵 ∣ ¬ 𝜑})
21uneq2d 3169 . 2 (𝐴 = 𝐵 → ({𝑥𝐶𝜑} ∪ {𝑥𝐴 ∣ ¬ 𝜑}) = ({𝑥𝐶𝜑} ∪ {𝑥𝐵 ∣ ¬ 𝜑}))
3 dfif6 3415 . 2 if(𝜑, 𝐶, 𝐴) = ({𝑥𝐶𝜑} ∪ {𝑥𝐴 ∣ ¬ 𝜑})
4 dfif6 3415 . 2 if(𝜑, 𝐶, 𝐵) = ({𝑥𝐶𝜑} ∪ {𝑥𝐵 ∣ ¬ 𝜑})
52, 3, 43eqtr4g 2152 1 (𝐴 = 𝐵 → if(𝜑, 𝐶, 𝐴) = if(𝜑, 𝐶, 𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1296  {crab 2374  cun 3011  ifcif 3413
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-rab 2379  df-v 2635  df-un 3017  df-if 3414
This theorem is referenced by:  ifeq12  3427  ifeq2d  3429  ifbieq2i  3434  xrmaxiflemcom  10808
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