Theorem bnj1351 34303

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

Hypothesis

Ref	Expression
bnj1351.1	⊢ (𝜑 → ∀𝑥𝜑)

Assertion

Ref	Expression
bnj1351	⊢ ((𝜑 ∧ 𝜓) → ∀𝑥(𝜑 ∧ 𝜓))

Distinct variable group:   𝜓,𝑥

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem bnj1351

Step	Hyp	Ref	Expression
1		bnj1351.1	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
2		ax-5 1912	. 2 ⊢ (𝜓 → ∀𝑥𝜓)
3	1, 2	hban 2295	1 ⊢ ((𝜑 ∧ 𝜓) → ∀𝑥(𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-10 2136  ax-12 2170

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-nf 1785

This theorem is referenced by:  bnj1373  34507  bnj1445  34521