Theorem pm2.61ne

Description: Deduction eliminating an inequality in an antecedent. (Contributed by NM, 24-May-2006) (Proof shortened by Andrew Salmon, 25-May-2011) (Proof shortened by Wolf Lammen, 25-Nov-2019)

	Ref	Expression
Hypotheses	pm2.61ne.1	⊢ ( 𝐴 = 𝐵 → ( 𝜓 ↔ 𝜒 ) )
	pm2.61ne.2	⊢ ( ( 𝜑 ∧ 𝐴 ≠ 𝐵 ) → 𝜓 )
	pm2.61ne.3	⊢ ( 𝜑 → 𝜒 )
Assertion	pm2.61ne	⊢ ( 𝜑 → 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		pm2.61ne.1	⊢ ( 𝐴 = 𝐵 → ( 𝜓 ↔ 𝜒 ) )
2		pm2.61ne.2	⊢ ( ( 𝜑 ∧ 𝐴 ≠ 𝐵 ) → 𝜓 )
3		pm2.61ne.3	⊢ ( 𝜑 → 𝜒 )
4	3 1	syl5ibr	⊢ ( 𝐴 = 𝐵 → ( 𝜑 → 𝜓 ) )
5	2	expcom	⊢ ( 𝐴 ≠ 𝐵 → ( 𝜑 → 𝜓 ) )
6	4 5	pm2.61ine	⊢ ( 𝜑 → 𝜓 )