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Theorem ifbi 4084
Description: Equivalence theorem for conditional operators. (Contributed by Raph Levien, 15-Jan-2004.)
Assertion
Ref Expression
ifbi ((𝜑𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))

Proof of Theorem ifbi
StepHypRef Expression
1 dfbi3 993 . 2 ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)))
2 iftrue 4069 . . . 4 (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
3 iftrue 4069 . . . . 5 (𝜓 → if(𝜓, 𝐴, 𝐵) = 𝐴)
43eqcomd 2627 . . . 4 (𝜓𝐴 = if(𝜓, 𝐴, 𝐵))
52, 4sylan9eq 2675 . . 3 ((𝜑𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))
6 iffalse 4072 . . . 4 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
7 iffalse 4072 . . . . 5 𝜓 → if(𝜓, 𝐴, 𝐵) = 𝐵)
87eqcomd 2627 . . . 4 𝜓𝐵 = if(𝜓, 𝐴, 𝐵))
96, 8sylan9eq 2675 . . 3 ((¬ 𝜑 ∧ ¬ 𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))
105, 9jaoi 394 . 2 (((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))
111, 10sylbi 207 1 ((𝜑𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384   = wceq 1480  ifcif 4063
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-if 4064
This theorem is referenced by:  ifbid  4085  ifbieq2i  4087  gsummoncoe1  19606  scmatscm  20251  mulmarep1gsum1  20311  madugsum  20381  mp2pm2mplem4  20546  dchrhash  24913  lgsdi  24976  rpvmasum2  25118  bj-projval  32666  matunitlindflem2  33073  itg2gt0cn  33132  dedths  33763
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