Theorem bnj1350 32805

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

Hypothesis

Ref	Expression
bnj1350.1	⊢ (𝜒 → ∀𝑥𝜒)

Assertion

Ref	Expression
bnj1350	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → ∀𝑥(𝜑 ∧ 𝜓 ∧ 𝜒))

Distinct variable groups:   𝜑,𝑥   𝜓,𝑥

Allowed substitution hint:   𝜒(𝑥)

Proof of Theorem bnj1350

Step	Hyp	Ref	Expression
1		ax-5 1913	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
2		ax-5 1913	. 2 ⊢ (𝜓 → ∀𝑥𝜓)
3		bnj1350.1	. 2 ⊢ (𝜒 → ∀𝑥𝜒)
4	1, 2, 3	hb3an 2298	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → ∀𝑥(𝜑 ∧ 𝜓 ∧ 𝜒))

Syntax hints:   → wi 4   ∧ w3a 1086  ∀wal 1537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-ex 1783  df-nf 1787

This theorem is referenced by:  bnj911  32912  bnj1093  32960