Theorem nimnbi 45168

Description: If an implication is false, the biconditional is false. (Contributed by Glauco Siliprandi, 15-Feb-2025.)

Hypothesis

Ref	Expression
nimnbi.1	⊢ ¬ (𝜑 → 𝜓)

Assertion

Ref	Expression
nimnbi	⊢ ¬ (𝜑 ↔ 𝜓)

Proof of Theorem nimnbi

Step	Hyp	Ref	Expression
1		nimnbi.1	. 2 ⊢ ¬ (𝜑 → 𝜓)
2		biimp 215	. 2 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓))
3	1, 2	mto 197	1 ⊢ ¬ (𝜑 ↔ 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206

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 207

This theorem is referenced by: (None)