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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 (φxφ)
hb.2 (ψxψ)
Assertion
Ref Expression
hban ((φ ψ) → x(φ ψ))

Proof of Theorem hban
StepHypRef Expression
1 hb.1 . . . 4 (φxφ)
21nfi 1551 . . 3 xφ
3 hb.2 . . . 4 (ψxψ)
43nfi 1551 . . 3 xψ
52, 4nfan 1824 . 2 x(φ ψ)
65nfri 1762 1 ((φ ψ) → x(φ ψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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  5562
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