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Theorem pm5.21ndd 343
 Description: Eliminate an antecedent implied by each side of a biconditional, deduction version. (Contributed by Paul Chapman, 21-Nov-2012.) (Proof shortened by Wolf Lammen, 6-Oct-2013.)
Hypotheses
Ref Expression
pm5.21ndd.1 (φ → (χψ))
pm5.21ndd.2 (φ → (θψ))
pm5.21ndd.3 (φ → (ψ → (χθ)))
Assertion
Ref Expression
pm5.21ndd (φ → (χθ))

Proof of Theorem pm5.21ndd
StepHypRef Expression
1 pm5.21ndd.3 . 2 (φ → (ψ → (χθ)))
2 pm5.21ndd.1 . . . 4 (φ → (χψ))
32con3d 125 . . 3 (φ → (¬ ψ → ¬ χ))
4 pm5.21ndd.2 . . . 4 (φ → (θψ))
54con3d 125 . . 3 (φ → (¬ ψ → ¬ θ))
6 pm5.21im 338 . . 3 χ → (¬ θ → (χθ)))
73, 5, 6syl6c 60 . 2 (φ → (¬ ψ → (χθ)))
81, 7pm2.61d 150 1 (φ → (χθ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8 This theorem depends on definitions:  df-bi 177 This theorem is referenced by:  pm5.21nd  868  rmob  3134  eqpw1uni  4330  fnasrn  5417  funiunfv  5467  eqncg  6126  eqtc  6161  elce  6175
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