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Metamath Proof Explorer

Theorem bnj1459

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

Ref Expression
Hypotheses bnj1459.1 ⊢ ( 𝜓 ↔ ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) )
bnj1459.2 ⊢ ( 𝜓 → 𝜒 )
Assertion bnj1459 ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝜒 )

Proof

Step Hyp Ref Expression
1 bnj1459.1 ⊢ ( 𝜓 ↔ ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) )
2 bnj1459.2 ⊢ ( 𝜓 → 𝜒 )
3 1 2 sylbir ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝜒 )
4 3 ralrimiva ⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝜒 )