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Metamath Proof Explorer

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