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Metamath Proof Explorer

Theorem necon4i

Description: Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007) (Proof shortened by Andrew Salmon, 25-May-2011) (Proof shortened by Wolf Lammen, 24-Nov-2019)

		Ref	Expression
	Hypothesis	necon4i.1	⊢ ( 𝐴 ≠ 𝐵 → 𝐶 ≠ 𝐷 )
	Assertion	necon4i	⊢ ( 𝐶 = 𝐷 → 𝐴 = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		necon4i.1	⊢ ( 𝐴 ≠ 𝐵 → 𝐶 ≠ 𝐷 )
2	1	neneqd	⊢ ( 𝐴 ≠ 𝐵 → ¬ 𝐶 = 𝐷 )
3	2	necon4ai	⊢ ( 𝐶 = 𝐷 → 𝐴 = 𝐵 )