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Theorem aifftbifffaibif 39737
 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 39713 . . . 4 𝜑
3 aifftbifffaibif.2 . . . . 5 (𝜓 ↔ ⊥)
43aisfina 39714 . . . 4 ¬ 𝜓
52, 4pm3.2i 470 . . 3 (𝜑 ∧ ¬ 𝜓)
6 annim 440 . . . 4 ((𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑𝜓))
76biimpi 205 . . 3 ((𝜑 ∧ ¬ 𝜓) → ¬ (𝜑𝜓))
85, 7ax-mp 5 . 2 ¬ (𝜑𝜓)
98bifal 1488 1 ((𝜑𝜓) ↔ ⊥)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383  ⊤wtru 1476  ⊥wfal 1480 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 196  df-an 385  df-tru 1478  df-fal 1481 This theorem is referenced by:  atnaiana  39739
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