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Theorem ifbi 3679
 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 863 . 2 ((φψ) ↔ ((φ ψ) φ ¬ ψ)))
2 iftrue 3668 . . . 4 (φ → if(φ, A, B) = A)
3 iftrue 3668 . . . . 5 (ψ → if(ψ, A, B) = A)
43eqcomd 2358 . . . 4 (ψA = if(ψ, A, B))
52, 4sylan9eq 2405 . . 3 ((φ ψ) → if(φ, A, B) = if(ψ, A, B))
6 iffalse 3669 . . . 4 φ → if(φ, A, B) = B)
7 iffalse 3669 . . . . 5 ψ → if(ψ, A, B) = B)
87eqcomd 2358 . . . 4 ψB = if(ψ, A, B))
96, 8sylan9eq 2405 . . 3 ((¬ φ ¬ ψ) → if(φ, A, B) = if(ψ, A, B))
105, 9jaoi 368 . 2 (((φ ψ) φ ¬ ψ)) → if(φ, A, B) = if(ψ, A, B))
111, 10sylbi 187 1 ((φψ) → if(φ, A, B) = if(ψ, A, B))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358   = wceq 1642   ifcif 3662 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-if 3663 This theorem is referenced by:  ifbid  3680  ifbieq2i  3681
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