Theorem abciffcbatnabciffncba

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)

	Ref	Expression
Assertion	abciffcbatnabciffncba	⊢ ( ¬ ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) → ¬ ( ( 𝜒 ∧ 𝜓 ) ∧ 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		an31	⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) ↔ ( ( 𝜒 ∧ 𝜓 ) ∧ 𝜑 ) )
2		notbi	⊢ ( ( ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) ↔ ( ( 𝜒 ∧ 𝜓 ) ∧ 𝜑 ) ) ↔ ( ¬ ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) ↔ ¬ ( ( 𝜒 ∧ 𝜓 ) ∧ 𝜑 ) ) )
3	2	biimpi	⊢ ( ( ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) ↔ ( ( 𝜒 ∧ 𝜓 ) ∧ 𝜑 ) ) → ( ¬ ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) ↔ ¬ ( ( 𝜒 ∧ 𝜓 ) ∧ 𝜑 ) ) )
4	1 3	ax-mp	⊢ ( ¬ ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) ↔ ¬ ( ( 𝜒 ∧ 𝜓 ) ∧ 𝜑 ) )
5	4	biimpi	⊢ ( ¬ ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) → ¬ ( ( 𝜒 ∧ 𝜓 ) ∧ 𝜑 ) )