Theorem bnj1352 35246

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

Hypothesis

Ref	Expression
bnj1352.1	⊢ (𝜓 → ∀𝑥𝜓)

Assertion

Ref	Expression
bnj1352	⊢ ((𝜑 ∧ 𝜓) → ∀𝑥(𝜑 ∧ 𝜓))

Distinct variable group:   𝜑,𝑥

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem bnj1352

Step	Hyp	Ref	Expression
1		ax-5 1943	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
2		bnj1352.1	. 2 ⊢ (𝜓 → ∀𝑥𝜓)
3	1, 2	hban 2338	1 ⊢ ((𝜑 ∧ 𝜓) → ∀𝑥(𝜑 ∧ 𝜓))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2179  ax-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817

This theorem is used by:  bnj594  35331  bnj1309  35441