Theorem bnj946 30099

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

Hypothesis

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

Assertion

Ref	Expression
bnj946	⊢ (𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))

Proof of Theorem bnj946

Step	Hyp	Ref	Expression
1		bnj946.1	. 2 ⊢ (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜓)
2		df-ral 2901	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))
3	1, 2	bitri 263	1 ⊢ (𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473   ∈ wcel 1977  ∀wral 2896

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 196  df-ral 2901

This theorem is referenced by:  bnj1379  30155  bnj570  30229  bnj571  30230