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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 )