Theorem bnj946 35405

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 3077	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))
3	1, 2	bitri 278	1 ⊢ (𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1567   ∈ wcel 2144  ∀wral 3076

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-ral 3077

This theorem is used by:  bnj1379  35460  bnj570  35535  bnj571  35536