Theorem bnj1095 34090

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 3296	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥∀𝑥 ∈ 𝐴 𝜓)
3	1, 2	hbxfrbi 1825	1 ⊢ (𝜑 → ∀𝑥𝜑)

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1537  ∀wral 3059

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-10 2135  ax-12 2169

This theorem depends on definitions:  df-bi 206  df-or 844  df-ex 1780  df-nf 1784  df-ral 3060

This theorem is referenced by:  bnj1379  34139  bnj605  34216  bnj594  34221  bnj607  34225  bnj911  34241  bnj964  34252  bnj983  34260  bnj1093  34289  bnj1123  34295  bnj1145  34302  bnj1417  34350