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

Theorem abciffcbatnabciffncba 43042
Description: Operands in a biconditional expression converted negated. Additionally biconditional converted to show antecedent implies sequent. Closed form. (Contributed by Jarvin Udandy, 7-Sep-2020.)
Assertion
Ref Expression
abciffcbatnabciffncba (¬ ((𝜑𝜓) ∧ 𝜒) → ¬ ((𝜒𝜓) ∧ 𝜑))

Proof of Theorem abciffcbatnabciffncba
StepHypRef Expression
1 an31 644 . . 3 (((𝜑𝜓) ∧ 𝜒) ↔ ((𝜒𝜓) ∧ 𝜑))
2 notbi 320 . . . 4 ((((𝜑𝜓) ∧ 𝜒) ↔ ((𝜒𝜓) ∧ 𝜑)) ↔ (¬ ((𝜑𝜓) ∧ 𝜒) ↔ ¬ ((𝜒𝜓) ∧ 𝜑)))
32biimpi 217 . . 3 ((((𝜑𝜓) ∧ 𝜒) ↔ ((𝜒𝜓) ∧ 𝜑)) → (¬ ((𝜑𝜓) ∧ 𝜒) ↔ ¬ ((𝜒𝜓) ∧ 𝜑)))
41, 3ax-mp 5 . 2 (¬ ((𝜑𝜓) ∧ 𝜒) ↔ ¬ ((𝜒𝜓) ∧ 𝜑))
54biimpi 217 1 (¬ ((𝜑𝜓) ∧ 𝜒) → ¬ ((𝜒𝜓) ∧ 𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396
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 208  df-an 397
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator