Theorem hban 1828

Description: If x is not free in φ and ψ, it is not free in (φ ∧ ψ). (Contributed by NM, 5-Aug-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	nfi 1551	. . 3 ⊢ Ⅎxφ
3		hb.2	. . . 4 ⊢ (ψ → ∀xψ)
4	3	nfi 1551	. . 3 ⊢ Ⅎxψ
5	2, 4	nfan 1824	. 2 ⊢ Ⅎx(φ ∧ ψ)
6	5	nfri 1762	1 ⊢ ((φ ∧ ψ) → ∀x(φ ∧ ψ))

Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by:  hb3anOLD  1831  dvelimv  1939  dvelimh  1964  dvelimALT  2133  dvelimf-o  2180  ax11indalem  2197  ax11inda2ALT  2198  cleqh  2450  hboprab  5563