Theorem bnj1095 30106

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 2926	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥∀𝑥 ∈ 𝐴 𝜓)
3	1, 2	hbxfrbi 1742	1 ⊢ (𝜑 → ∀𝑥𝜑)

Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473  ∀wral 2896

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034

This theorem depends on definitions:  df-bi 196  df-or 384  df-ex 1696  df-nf 1701  df-ral 2901

This theorem is referenced by:  bnj1379  30155  bnj605  30231  bnj594  30236  bnj607  30240  bnj911  30256  bnj964  30267  bnj983  30275  bnj1093  30302  bnj1123  30308  bnj1145  30315  bnj1417  30363