Theorem imnang

Description: Quantified implication in terms of quantified negation of conjunction. (Contributed by BJ, 16-Jul-2021)

	Ref	Expression
Assertion	imnang	⊢ ( ∀ 𝑥 ( 𝜑 → ¬ 𝜓 ) ↔ ∀ 𝑥 ¬ ( 𝜑 ∧ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		imnan	⊢ ( ( 𝜑 → ¬ 𝜓 ) ↔ ¬ ( 𝜑 ∧ 𝜓 ) )
2	1	albii	⊢ ( ∀ 𝑥 ( 𝜑 → ¬ 𝜓 ) ↔ ∀ 𝑥 ¬ ( 𝜑 ∧ 𝜓 ) )