Theorem bnj1350 35455

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 1942	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
2		ax-5 1942	. 2 ⊢ (𝜓 → ∀𝑥𝜓)
3		bnj1350.1	. 2 ⊢ (𝜒 → ∀𝑥𝜒)
4	1, 2, 3	hb3an 2334	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → ∀𝑥(𝜑 ∧ 𝜓 ∧ 𝜒))

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103  ∀wal 1567

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-10 2177  ax-12 2212

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-ex 1812  df-nf 1816

This theorem is used by:  bnj911  35562  bnj1093  35610