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Metamath Proof Explorer

Theorem pm2.61ine

Description: Inference eliminating an inequality in an antecedent. (Contributed by NM, 16-Jan-2007) (Proof shortened by Andrew Salmon, 25-May-2011)

	Ref	Expression
Hypotheses	pm2.61ine.1	⊢ ( 𝐴 = 𝐵 → 𝜑 )
	pm2.61ine.2	⊢ ( 𝐴 ≠ 𝐵 → 𝜑 )
Assertion	pm2.61ine	⊢ 𝜑

Proof

Step	Hyp	Ref	Expression
1		pm2.61ine.1	⊢ ( 𝐴 = 𝐵 → 𝜑 )
2		pm2.61ine.2	⊢ ( 𝐴 ≠ 𝐵 → 𝜑 )
3		nne	⊢ ( ¬ 𝐴 ≠ 𝐵 ↔ 𝐴 = 𝐵 )
4	3 1	sylbi	⊢ ( ¬ 𝐴 ≠ 𝐵 → 𝜑 )
5	2 4	pm2.61i	⊢ 𝜑