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Theorem aifftbifffaibif 41602
Description: Given a is equivalent to T., Given b is equivalent to F., there exists a proof for that a implies b is false. (Contributed by Jarvin Udandy, 7-Sep-2020.)
Hypotheses
Ref Expression
aifftbifffaibif.1 (𝜑 ↔ ⊤)
aifftbifffaibif.2 (𝜓 ↔ ⊥)
Assertion
Ref Expression
aifftbifffaibif ((𝜑𝜓) ↔ ⊥)

Proof of Theorem aifftbifffaibif
StepHypRef Expression
1 aifftbifffaibif.1 . . . . 5 (𝜑 ↔ ⊤)
21aistia 41578 . . . 4 𝜑
3 aifftbifffaibif.2 . . . . 5 (𝜓 ↔ ⊥)
43aisfina 41579 . . . 4 ¬ 𝜓
52, 4pm3.2i 447 . . 3 (𝜑 ∧ ¬ 𝜓)
6 annim 390 . . . 4 ((𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑𝜓))
76biimpi 206 . . 3 ((𝜑 ∧ ¬ 𝜓) → ¬ (𝜑𝜓))
85, 7ax-mp 5 . 2 ¬ (𝜑𝜓)
98bifal 1644 1 ((𝜑𝜓) ↔ ⊥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  wtru 1631  wfal 1635
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  df-an 383  df-tru 1633  df-fal 1636
This theorem is referenced by:  atnaiana  41604
  Copyright terms: Public domain W3C validator