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Theorem elimif 3691
 Description: Elimination of a conditional operator contained in a wff ψ. (Contributed by NM, 15-Feb-2005.)
Hypotheses
Ref Expression
elimif.1 ( if(φ, A, B) = A → (ψχ))
elimif.2 ( if(φ, A, B) = B → (ψθ))
Assertion
Ref Expression
elimif (ψ ↔ ((φ χ) φ θ)))

Proof of Theorem elimif
StepHypRef Expression
1 exmid 404 . . 3 (φ ¬ φ)
21biantrur 492 . 2 (ψ ↔ ((φ ¬ φ) ψ))
3 andir 838 . 2 (((φ ¬ φ) ψ) ↔ ((φ ψ) φ ψ)))
4 iftrue 3668 . . . . 5 (φ → if(φ, A, B) = A)
5 elimif.1 . . . . 5 ( if(φ, A, B) = A → (ψχ))
64, 5syl 15 . . . 4 (φ → (ψχ))
76pm5.32i 618 . . 3 ((φ ψ) ↔ (φ χ))
8 iffalse 3669 . . . . 5 φ → if(φ, A, B) = B)
9 elimif.2 . . . . 5 ( if(φ, A, B) = B → (ψθ))
108, 9syl 15 . . . 4 φ → (ψθ))
1110pm5.32i 618 . . 3 ((¬ φ ψ) ↔ (¬ φ θ))
127, 11orbi12i 507 . 2 (((φ ψ) φ ψ)) ↔ ((φ χ) φ θ)))
132, 3, 123bitri 262 1 (ψ ↔ ((φ χ) φ θ)))
 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:  eqif  3695  elif  3696  ifel  3697
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