Theorem bnj1095 34979

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

Hypothesis

Ref	Expression
bnj1095.1	⊢ (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜓)

Assertion

Ref	Expression
bnj1095	⊢ (𝜑 → ∀𝑥𝜑)

Proof of Theorem bnj1095

Step	Hyp	Ref	Expression
1		bnj1095.1	. 2 ⊢ (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜓)
2		hbra1 3278	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥∀𝑥 ∈ 𝐴 𝜓)
3	1, 2	hbxfrbi 1833	1 ⊢ (𝜑 → ∀𝑥𝜑)

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1546  ∀wral 3055

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-10 2154  ax-12 2191

This theorem depends on definitions:  df-bi 209  df-or 855  df-ex 1788  df-nf 1792  df-ral 3056

This theorem is referenced by:  bnj1379  35027  bnj605  35104  bnj594  35109  bnj607  35113  bnj911  35129  bnj964  35140  bnj983  35148  bnj1093  35177  bnj1123  35183  bnj1145  35190  bnj1417  35238