Theorem bnj1211 35194

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

Hypothesis

Ref	Expression
bnj1211.1	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)

Assertion

Ref	Expression
bnj1211	⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))

Proof of Theorem bnj1211

Step	Hyp	Ref	Expression
1		bnj1211.1	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)
2		df-ral 3079	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))
3	1, 2	sylib 221	1 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))

This proof depends on syntax axioms:   → wi 4  ∀wal 1567   ∈ wcel 2142  ∀wral 3078

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 3079

This theorem is used by:  bnj1533  35249  bnj1204  35409  bnj1523  35468