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Metamath Proof Explorer

Theorem bnj946

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

Ref Expression
Hypothesis bnj946.1 ⊢ ( 𝜑 ↔ ∀ 𝑥 ∈ 𝐴 𝜓 )
Assertion bnj946 ( 𝜑 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜓 ) )

Proof

Step Hyp Ref Expression
1 bnj946.1 ⊢ ( 𝜑 ↔ ∀ 𝑥 ∈ 𝐴 𝜓 )
2 df-ral ⊢ ( ∀ 𝑥 ∈ 𝐴 𝜓 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜓 ) )
3 1 2 bitri ⊢ ( 𝜑 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜓 ) )