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Theorem ifclda 3689
 Description: Conditional closure. (Contributed by Jeff Madsen, 2-Sep-2009.)
Hypotheses
Ref Expression
ifclda.1 ((φ ψ) → A C)
ifclda.2 ((φ ¬ ψ) → B C)
Assertion
Ref Expression
ifclda (φ → if(ψ, A, B) C)

Proof of Theorem ifclda
StepHypRef Expression
1 iftrue 3668 . . . 4 (ψ → if(ψ, A, B) = A)
21adantl 452 . . 3 ((φ ψ) → if(ψ, A, B) = A)
3 ifclda.1 . . 3 ((φ ψ) → A C)
42, 3eqeltrd 2427 . 2 ((φ ψ) → if(ψ, A, B) C)
5 iffalse 3669 . . . 4 ψ → if(ψ, A, B) = B)
65adantl 452 . . 3 ((φ ¬ ψ) → if(ψ, A, B) = B)
7 ifclda.2 . . 3 ((φ ¬ ψ) → B C)
86, 7eqeltrd 2427 . 2 ((φ ¬ ψ) → if(ψ, A, B) C)
94, 8pm2.61dan 766 1 (φ → if(ψ, A, B) C)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710   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: (None)
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