Theorem hban 2338

Description: If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑 ∧ 𝜓). (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)

Hypotheses

Ref	Expression
hb.1	⊢ (𝜑 → ∀𝑥𝜑)
hb.2	⊢ (𝜓 → ∀𝑥𝜓)

Assertion

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

Proof of Theorem hban

Step	Hyp	Ref	Expression
1		hb.1	. . . 4 ⊢ (𝜑 → ∀𝑥𝜑)
2	1	nf5i 2184	. . 3 ⊢ Ⅎ𝑥𝜑
3		hb.2	. . . 4 ⊢ (𝜓 → ∀𝑥𝜓)
4	3	nf5i 2184	. . 3 ⊢ Ⅎ𝑥𝜓
5	2, 4	nfan 1932	. 2 ⊢ Ⅎ𝑥(𝜑 ∧ 𝜓)
6	5	nf5ri 2234	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:  bnj982  35199  bnj1351  35246  bnj1352  35247  bnj1441  35260  bnj1441g  35261  copsex2b  37825  dvelimf-o  39744  ax12indalem  39760  ax12inda2ALT  39761  hbimpg  45304  hbimpgVD  45653