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Theorem abciffcbatnabciffncbai 41611
Description: Operands in a biconditional expression converted negated. Additionally biconditional converted to show antecedent implies sequent. (Contributed by Jarvin Udandy, 7-Sep-2020.)
Hypothesis
Ref Expression
abciffcbatnabciffncbai.1 (((𝜑𝜓) ∧ 𝜒) ↔ ((𝜒𝜓) ∧ 𝜑))
Assertion
Ref Expression
abciffcbatnabciffncbai (¬ ((𝜑𝜓) ∧ 𝜒) → ¬ ((𝜒𝜓) ∧ 𝜑))

Proof of Theorem abciffcbatnabciffncbai
StepHypRef Expression
1 abciffcbatnabciffncbai.1 . . 3 (((𝜑𝜓) ∧ 𝜒) ↔ ((𝜒𝜓) ∧ 𝜑))
2 notbi 308 . . . 4 ((((𝜑𝜓) ∧ 𝜒) ↔ ((𝜒𝜓) ∧ 𝜑)) ↔ (¬ ((𝜑𝜓) ∧ 𝜒) ↔ ¬ ((𝜒𝜓) ∧ 𝜑)))
32biimpi 206 . . 3 ((((𝜑𝜓) ∧ 𝜒) ↔ ((𝜒𝜓) ∧ 𝜑)) → (¬ ((𝜑𝜓) ∧ 𝜒) ↔ ¬ ((𝜒𝜓) ∧ 𝜑)))
41, 3ax-mp 5 . 2 (¬ ((𝜑𝜓) ∧ 𝜒) ↔ ¬ ((𝜒𝜓) ∧ 𝜑))
54biimpi 206 1 (¬ ((𝜑𝜓) ∧ 𝜒) → ¬ ((𝜒𝜓) ∧ 𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382
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
This theorem is referenced by: (None)
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