Theorem pm2.61dne

Description: Deduction eliminating an inequality in an antecedent. (Contributed by NM, 1-Jun-2007) (Proof shortened by Andrew Salmon, 25-May-2011)

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

Proof

Step	Hyp	Ref	Expression
1		pm2.61dne.1	⊢ ( 𝜑 → ( 𝐴 = 𝐵 → 𝜓 ) )
2		pm2.61dne.2	⊢ ( 𝜑 → ( 𝐴 ≠ 𝐵 → 𝜓 ) )
3		nne	⊢ ( ¬ 𝐴 ≠ 𝐵 ↔ 𝐴 = 𝐵 )
4	3 1	syl5bi	⊢ ( 𝜑 → ( ¬ 𝐴 ≠ 𝐵 → 𝜓 ) )
5	2 4	pm2.61d	⊢ ( 𝜑 → 𝜓 )