Theorem hban 2297

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 2142	. . 3 ⊢ Ⅎ𝑥𝜑
3		hb.2	. . . 4 ⊢ (𝜓 → ∀𝑥𝜓)
4	3	nf5i 2142	. . 3 ⊢ Ⅎ𝑥𝜓
5	2, 4	nfan 1902	. 2 ⊢ Ⅎ𝑥(𝜑 ∧ 𝜓)
6	5	nf5ri 2188	1 ⊢ ((𝜑 ∧ 𝜓) → ∀𝑥(𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 396  ∀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-tru 1542  df-ex 1783  df-nf 1787

This theorem is referenced by:  bnj982  32758  bnj1351  32806  bnj1352  32807  bnj1441  32820  bnj1441g  32821  copsex2b  35311  dvelimf-o  36943  ax12indalem  36959  ax12inda2ALT  36960  hbimpg  42174  hbimpgVD  42524