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Theorem pm5.21ndd 620
Description: Eliminate an antecedent implied by each side of a biconditional, deduction version. (Contributed by Paul Chapman, 21-Nov-2012.) (Revised by Mario Carneiro, 31-Jan-2015.)
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.1 . . . 4 (φ → (χψ))
2 pm5.21ndd.3 . . . 4 (φ → (ψ → (χθ)))
31, 2syld 40 . . 3 (φ → (χ → (χθ)))
43ibd 167 . 2 (φ → (χθ))
5 pm5.21ndd.2 . . . . 5 (φ → (θψ))
65, 2syld 40 . . . 4 (φ → (θ → (χθ)))
7 bicom1 122 . . . 4 ((χθ) → (θχ))
86, 7syl6 29 . . 3 (φ → (θ → (θχ)))
98ibd 167 . 2 (φ → (θχ))
104, 9impbid 120 1 (φ → (χθ))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101
This theorem depends on definitions:  df-bi 110
This theorem is referenced by:  pm5.21nd  824  rmob  2844  epelg  4018  eqbrrdva  4448  relbrcnvg  4647  fmptco  5273  ovelrn  5591  brtpos2  5807  brdomg  6165  genpelvl  6494  genpelvu  6495  fzoval  8735
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