Theorem pm2.61danel 3070

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

Hypotheses

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

Assertion

Ref	Expression
pm2.61danel	⊢ (𝜑 → 𝜓)

Proof of Theorem pm2.61danel

Step	Hyp	Ref	Expression
1		pm2.61danel.1	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝐵) → 𝜓)
2		df-nel 3057	. . 3 ⊢ (𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵)
3		pm2.61danel.2	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∉ 𝐵) → 𝜓)
4	2, 3	sylan2br 598	. 2 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ 𝐵) → 𝜓)
5	1, 4	pm2.61dan 813	1 ⊢ (𝜑 → 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   ∈ wcel 2112   ∉ wnel 3056

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 210  df-an 401  df-nel 3057

This theorem is referenced by:  clwwlknon1le1  27986  nsnlpligALT  28365  n0lpligALT  28367