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Theorem ifbi 4251
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 1036 . 2 ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)))
2 iftrue 4236 . . . 4 (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
3 iftrue 4236 . . . . 5 (𝜓 → if(𝜓, 𝐴, 𝐵) = 𝐴)
43eqcomd 2766 . . . 4 (𝜓𝐴 = if(𝜓, 𝐴, 𝐵))
52, 4sylan9eq 2814 . . 3 ((𝜑𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))
6 iffalse 4239 . . . 4 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
7 iffalse 4239 . . . . 5 𝜓 → if(𝜓, 𝐴, 𝐵) = 𝐵)
87eqcomd 2766 . . . 4 𝜓𝐵 = if(𝜓, 𝐴, 𝐵))
96, 8sylan9eq 2814 . . 3 ((¬ 𝜑 ∧ ¬ 𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))
105, 9jaoi 393 . 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 382  wa 383   = wceq 1632  ifcif 4230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-if 4231
This theorem is referenced by:  ifbid  4252  ifbieq2i  4254  gsummoncoe1  19896  scmatscm  20541  mulmarep1gsum1  20601  madugsum  20671  mp2pm2mplem4  20836  dchrhash  25216  lgsdi  25279  rpvmasum2  25421  bj-projval  33308  matunitlindflem2  33737  itg2gt0cn  33796  dedths  34769
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