Theorem pm2.61dane

Description: Deduction eliminating an inequality in an antecedent. (Contributed by NM, 30-Nov-2011)

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

Proof

Step	Hyp	Ref	Expression
1		pm2.61dane.1	⊢ ( ( 𝜑 ∧ 𝐴 = 𝐵 ) → 𝜓 )
2		pm2.61dane.2	⊢ ( ( 𝜑 ∧ 𝐴 ≠ 𝐵 ) → 𝜓 )
3	1	ex	⊢ ( 𝜑 → ( 𝐴 = 𝐵 → 𝜓 ) )
4	2	ex	⊢ ( 𝜑 → ( 𝐴 ≠ 𝐵 → 𝜓 ) )
5	3 4	pm2.61dne	⊢ ( 𝜑 → 𝜓 )