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Metamath Proof Explorer

Theorem necon1bd

Description: Contrapositive deduction for inequality. (Contributed by NM, 21-Mar-2007) (Proof shortened by Andrew Salmon, 25-May-2011) (Proof shortened by Wolf Lammen, 23-Nov-2019)

		Ref	Expression
	Hypothesis	necon1bd.1	⊢ ( 𝜑 → ( 𝐴 ≠ 𝐵 → 𝜓 ) )
	Assertion	necon1bd	⊢ ( 𝜑 → ( ¬ 𝜓 → 𝐴 = 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		necon1bd.1	⊢ ( 𝜑 → ( 𝐴 ≠ 𝐵 → 𝜓 ) )
2		df-ne	⊢ ( 𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵 )
3	2 1	biimtrrid	⊢ ( 𝜑 → ( ¬ 𝐴 = 𝐵 → 𝜓 ) )
4	3	con1d	⊢ ( 𝜑 → ( ¬ 𝜓 → 𝐴 = 𝐵 ) )