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Metamath Proof Explorer

Theorem necon4bd

Description: Contrapositive inference for inequality. (Contributed by NM, 1-Jun-2007) (Proof shortened by Andrew Salmon, 25-May-2011) (Proof shortened by Wolf Lammen, 23-Nov-2019)

		Ref	Expression
	Hypothesis	necon4bd.1	⊢ ( 𝜑 → ( ¬ 𝜓 → 𝐴 ≠ 𝐵 ) )
	Assertion	necon4bd	⊢ ( 𝜑 → ( 𝐴 = 𝐵 → 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		necon4bd.1	⊢ ( 𝜑 → ( ¬ 𝜓 → 𝐴 ≠ 𝐵 ) )
2	1	necon2bd	⊢ ( 𝜑 → ( 𝐴 = 𝐵 → ¬ ¬ 𝜓 ) )
3		notnotr	⊢ ( ¬ ¬ 𝜓 → 𝜓 )
4	2 3	syl6	⊢ ( 𝜑 → ( 𝐴 = 𝐵 → 𝜓 ) )