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Theorem mtt 643
Description: Modus-tollens-like theorem. (Contributed by NM, 7-Apr-2001.) (Revised by Mario Carneiro, 31-Jan-2015.)
Assertion
Ref Expression
mtt 𝜑 → (¬ 𝜓 ↔ (𝜓𝜑)))

Proof of Theorem mtt
StepHypRef Expression
1 pm2.21 580 . 2 𝜓 → (𝜓𝜑))
2 con3 604 . . 3 ((𝜓𝜑) → (¬ 𝜑 → ¬ 𝜓))
32com12 30 . 2 𝜑 → ((𝜓𝜑) → ¬ 𝜓))
41, 3impbid2 141 1 𝜑 → (¬ 𝜓 ↔ (𝜓𝜑)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 103
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  nbn2  646  dfnot  1305
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