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 ) )