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Metamath Proof Explorer

Theorem necon4ai

Description: Contrapositive inference for inequality. (Contributed by NM, 16-Jan-2007) (Proof shortened by Andrew Salmon, 25-May-2011) (Proof shortened by Wolf Lammen, 22-Nov-2019)

		Ref	Expression
	Hypothesis	necon4ai.1	⊢ ( 𝐴 ≠ 𝐵 → ¬ 𝜑 )
	Assertion	necon4ai	⊢ ( 𝜑 → 𝐴 = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		necon4ai.1	⊢ ( 𝐴 ≠ 𝐵 → ¬ 𝜑 )
2		notnot	⊢ ( 𝜑 → ¬ ¬ 𝜑 )
3	1	necon1bi	⊢ ( ¬ ¬ 𝜑 → 𝐴 = 𝐵 )
4	2 3	syl	⊢ ( 𝜑 → 𝐴 = 𝐵 )