Theorem bnj1095 35179

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 3301	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥∀𝑥 ∈ 𝐴 𝜓)
3	1, 2	hbxfrbi 1854	1 ⊢ (𝜑 → ∀𝑥𝜑)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1567  ∀wral 3078

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-10 2175  ax-12 2212

This proof depends on definitions:  df-bi 210  df-or 861  df-ex 1809  df-nf 1813  df-ral 3079

This theorem is used by:  bnj1379  35227  bnj605  35304  bnj594  35309  bnj607  35313  bnj911  35329  bnj964  35340  bnj983  35348  bnj1093  35377  bnj1123  35383  bnj1145  35390  bnj1417  35438