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Theorem ifbi 4486
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 1043 . 2 ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)))
2 iftrue 4471 . . . 4 (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
3 iftrue 4471 . . . . 5 (𝜓 → if(𝜓, 𝐴, 𝐵) = 𝐴)
43eqcomd 2825 . . . 4 (𝜓𝐴 = if(𝜓, 𝐴, 𝐵))
52, 4sylan9eq 2874 . . 3 ((𝜑𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))
6 iffalse 4474 . . . 4 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
7 iffalse 4474 . . . . 5 𝜓 → if(𝜓, 𝐴, 𝐵) = 𝐵)
87eqcomd 2825 . . . 4 𝜓𝐵 = if(𝜓, 𝐴, 𝐵))
96, 8sylan9eq 2874 . . 3 ((¬ 𝜑 ∧ ¬ 𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))
105, 9jaoi 853 . 2 (((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))
111, 10sylbi 219 1 ((𝜑𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  wo 843   = wceq 1530  ifcif 4465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1774  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-if 4466
This theorem is referenced by:  ifbid  4487  ifbieq2i  4489  gsummoncoe1  20464  scmatscm  21114  mulmarep1gsum1  21174  madugsum  21244  mp2pm2mplem4  21409  dchrhash  25839  lgsdi  25902  rpvmasum2  26080  bj-projval  34301  matunitlindflem2  34881  itg2gt0cn  34939  dedths  36090  dfafv2  43321
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