Theorem bnj1095 32163

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 3184	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥∀𝑥 ∈ 𝐴 𝜓)
3	1, 2	hbxfrbi 1826	1 ⊢ (𝜑 → ∀𝑥𝜑)

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536  ∀wral 3106

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-12 2175

This theorem depends on definitions:  df-bi 210  df-or 845  df-ex 1782  df-nf 1786  df-ral 3111

This theorem is referenced by:  bnj1379  32212  bnj605  32289  bnj594  32294  bnj607  32298  bnj911  32314  bnj964  32325  bnj983  32333  bnj1093  32362  bnj1123  32368  bnj1145  32375  bnj1417  32423