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Metamath Proof Explorer

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 ) )