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Metamath Proof Explorer

Theorem necon3ad

Description: Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007) (Proof shortened by Andrew Salmon, 25-May-2011) (Proof shortened by Wolf Lammen, 23-Nov-2019)

		Ref	Expression
	Hypothesis	necon3ad.1	⊢ ( 𝜑 → ( 𝜓 → 𝐴 = 𝐵 ) )
	Assertion	necon3ad	⊢ ( 𝜑 → ( 𝐴 ≠ 𝐵 → ¬ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		necon3ad.1	⊢ ( 𝜑 → ( 𝜓 → 𝐴 = 𝐵 ) )
2		neneq	⊢ ( 𝐴 ≠ 𝐵 → ¬ 𝐴 = 𝐵 )
3	1 2	nsyli	⊢ ( 𝜑 → ( 𝐴 ≠ 𝐵 → ¬ 𝜓 ) )