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Theorem ifbi 4512
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 1049 . 2 ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)))
2 iftrue 4496 . . . 4 (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
3 iftrue 4496 . . . . 5 (𝜓 → if(𝜓, 𝐴, 𝐵) = 𝐴)
43eqcomd 2739 . . . 4 (𝜓𝐴 = if(𝜓, 𝐴, 𝐵))
52, 4sylan9eq 2793 . . 3 ((𝜑𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))
6 iffalse 4499 . . . 4 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
7 iffalse 4499 . . . . 5 𝜓 → if(𝜓, 𝐴, 𝐵) = 𝐵)
87eqcomd 2739 . . . 4 𝜓𝐵 = if(𝜓, 𝐴, 𝐵))
96, 8sylan9eq 2793 . . 3 ((¬ 𝜑 ∧ ¬ 𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))
105, 9jaoi 856 . 2 (((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))
111, 10sylbi 216 1 ((𝜑𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wo 846   = wceq 1542  ifcif 4490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-if 4491
This theorem is referenced by:  ifbid  4513  ifbieq2i  4515  gsummoncoe1  21698  scmatscm  21885  mulmarep1gsum1  21945  madugsum  22015  mp2pm2mplem4  22181  dchrhash  26642  lgsdi  26705  rpvmasum2  26883  ifnebib  31521  bj-projval  35517  matunitlindflem2  36125  itg2gt0cn  36183  dedths  37474  dfafv2  45454
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