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Metamath Proof Explorer

Theorem imnot

Description: If a proposition is false, then implying it is equivalent to being false. One of four theorems that can be used to simplify an implication ( ph -> ps ) , the other ones being ax-1 (true consequent), pm2.21 (false antecedent), pm5.5 (true antecedent). (Contributed by Mario Carneiro, 26-Apr-2019) (Proof shortened by Wolf Lammen, 26-May-2019)

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

Proof

Step	Hyp	Ref	Expression
1		mtt	⊢ ( ¬ 𝜓 → ( ¬ 𝜑 ↔ ( 𝜑 → 𝜓 ) ) )
2	1	bicomd	⊢ ( ¬ 𝜓 → ( ( 𝜑 → 𝜓 ) ↔ ¬ 𝜑 ) )