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Theorem bija 369
Description: Combine antecedents into a single biconditional. This inference, reminiscent of ja 173, is reversible: The hypotheses can be deduced from the conclusion alone (see pm5.1im 253 and pm5.21im 363). (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 biimpr 210 . . 3 ((𝜑𝜓) → (𝜓𝜑))
2 bija.1 . . 3 (𝜑 → (𝜓𝜒))
31, 2syli 39 . 2 ((𝜑𝜓) → (𝜓𝜒))
4 biimp 205 . . . 4 ((𝜑𝜓) → (𝜑𝜓))
54con3d 148 . . 3 ((𝜑𝜓) → (¬ 𝜓 → ¬ 𝜑))
6 bija.2 . . 3 𝜑 → (¬ 𝜓𝜒))
75, 6syli 39 . 2 ((𝜑𝜓) → (¬ 𝜓𝜒))
83, 7pm2.61d 170 1 ((𝜑𝜓) → 𝜒)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196
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 197
This theorem is referenced by:  equvel  2375  2lgsoddprm  25186  bj-bibibi  32696  wl-aleq  33452  wl-nfeqfb  33453  rp-fakeimass  38174  rp-fakenanass  38177
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