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Metamath Proof Explorer

Theorem nbn2

Description: The negation of a wff is equivalent to the wff's equivalence to falsehood. (Contributed by Juha Arpiainen, 19-Jan-2006) (Proof shortened by Wolf Lammen, 28-Jan-2013)

		Ref	Expression
	Assertion	nbn2	⊢ ( ¬ 𝜑 → ( ¬ 𝜓 ↔ ( 𝜑 ↔ 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		pm5.501	⊢ ( ¬ 𝜑 → ( ¬ 𝜓 ↔ ( ¬ 𝜑 ↔ ¬ 𝜓 ) ) )
2		notbi	⊢ ( ( 𝜑 ↔ 𝜓 ) ↔ ( ¬ 𝜑 ↔ ¬ 𝜓 ) )
3	1 2	bitr4di	⊢ ( ¬ 𝜑 → ( ¬ 𝜓 ↔ ( 𝜑 ↔ 𝜓 ) ) )