Users' Mathboxes Mathbox for Jarvin Udandy < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  aifftbifffaibifff Structured version   Visualization version   GIF version

Theorem aifftbifffaibifff 41410
Description: Given a is equivalent to T., Given b is equivalent to F., there exists a proof for that a iff b is false. (Contributed by Jarvin Udandy, 7-Sep-2020.)
Hypotheses
Ref Expression
aifftbifffaibifff.1 (𝜑 ↔ ⊤)
aifftbifffaibifff.2 (𝜓 ↔ ⊥)
Assertion
Ref Expression
aifftbifffaibifff ((𝜑𝜓) ↔ ⊥)

Proof of Theorem aifftbifffaibifff
StepHypRef Expression
1 aifftbifffaibifff.1 . . . . 5 (𝜑 ↔ ⊤)
21aistia 41385 . . . 4 𝜑
3 aifftbifffaibifff.2 . . . . 5 (𝜓 ↔ ⊥)
43aisfina 41386 . . . 4 ¬ 𝜓
52, 4abnotbtaxb 41403 . . 3 (𝜑𝜓)
65axorbtnotaiffb 41391 . 2 ¬ (𝜑𝜓)
7 nbfal 1535 . . 3 (¬ (𝜑𝜓) ↔ ((𝜑𝜓) ↔ ⊥))
87biimpi 206 . 2 (¬ (𝜑𝜓) → ((𝜑𝜓) ↔ ⊥))
96, 8ax-mp 5 1 ((𝜑𝜓) ↔ ⊥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wtru 1524  wfal 1528
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 385  df-xor 1505  df-tru 1526  df-fal 1529
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator