Theorem pm2.61danel

Description: Deduction eliminating an elementhood in an antecedent. (Contributed by AV, 5-Dec-2021)

	Ref	Expression
Hypotheses	pm2.61danel.1	⊢ ( ( 𝜑 ∧ 𝐴 ∈ 𝐵 ) → 𝜓 )
	pm2.61danel.2	⊢ ( ( 𝜑 ∧ 𝐴 ∉ 𝐵 ) → 𝜓 )
Assertion	pm2.61danel	⊢ ( 𝜑 → 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		pm2.61danel.1	⊢ ( ( 𝜑 ∧ 𝐴 ∈ 𝐵 ) → 𝜓 )
2		pm2.61danel.2	⊢ ( ( 𝜑 ∧ 𝐴 ∉ 𝐵 ) → 𝜓 )
3		df-nel	⊢ ( 𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵 )
4	3 2	sylan2br	⊢ ( ( 𝜑 ∧ ¬ 𝐴 ∈ 𝐵 ) → 𝜓 )
5	1 4	pm2.61dan	⊢ ( 𝜑 → 𝜓 )