Theorem bnj1276 32694

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

Hypotheses

Ref	Expression
bnj1276.1	⊢ (𝜑 → ∀𝑥𝜑)
bnj1276.2	⊢ (𝜓 → ∀𝑥𝜓)
bnj1276.3	⊢ (𝜒 → ∀𝑥𝜒)
bnj1276.4	⊢ (𝜃 ↔ (𝜑 ∧ 𝜓 ∧ 𝜒))

Assertion

Ref	Expression
bnj1276	⊢ (𝜃 → ∀𝑥𝜃)

Proof of Theorem bnj1276

Step	Hyp	Ref	Expression
1		bnj1276.4	. 2 ⊢ (𝜃 ↔ (𝜑 ∧ 𝜓 ∧ 𝜒))
2		bnj1276.1	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
3		bnj1276.2	. . 3 ⊢ (𝜓 → ∀𝑥𝜓)
4		bnj1276.3	. . 3 ⊢ (𝜒 → ∀𝑥𝜒)
5	2, 3, 4	hb3an 2301	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → ∀𝑥(𝜑 ∧ 𝜓 ∧ 𝜒))
6	1, 5	hbxfrbi 1828	1 ⊢ (𝜃 → ∀𝑥𝜃)

Syntax hints:   → wi 4   ↔ wb 205   ∧ w3a 1085  ∀wal 1537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2139  ax-12 2173

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-ex 1784  df-nf 1788

This theorem is referenced by: (None)