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Theorem bija 344
 Description: Combine antecedents into a single bi-conditional. This inference, reminiscent of ja 153, is reversible: The hypotheses can be deduced from the conclusion alone (see pm5.1im 229 and pm5.21im 338). (Contributed by Wolf Lammen, 13-May-2013.)
Hypotheses
Ref Expression
bija.1 (φ → (ψχ))
bija.2 φ → (¬ ψχ))
Assertion
Ref Expression
bija ((φψ) → χ)

Proof of Theorem bija
StepHypRef Expression
1 bi2 189 . . 3 ((φψ) → (ψφ))
2 bija.1 . . 3 (φ → (ψχ))
31, 2syli 33 . 2 ((φψ) → (ψχ))
4 bi1 178 . . . 4 ((φψ) → (φψ))
54con3d 125 . . 3 ((φψ) → (¬ ψ → ¬ φ))
6 bija.2 . . 3 φ → (¬ ψχ))
75, 6syli 33 . 2 ((φψ) → (¬ ψχ))
83, 7pm2.61d 150 1 ((φψ) → χ)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8 This theorem depends on definitions:  df-bi 177 This theorem is referenced by: (None)
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