Theorem hban 2128

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

Hypotheses

Ref	Expression
hb.1	|- ( ph -> A. x ph )
hb.2	|- ( ps -> A. x ps )

Assertion

Ref	Expression
hban	|- ( ( ph /\ ps ) -> A. x ( ph /\ ps ) )

Proof of Theorem hban

Step	Hyp	Ref	Expression
1		hb.1	. . . 4 |- ( ph -> A. x ph )
2	1	nf5i 2024	. . 3 |- F/ x ph
3		hb.2	. . . 4 |- ( ps -> A. x ps )
4	3	nf5i 2024	. . 3 |- F/ x ps
5	2, 4	nfan 1828	. 2 |- F/ x ( ph /\ ps )
6	5	nf5ri 2065	1 |- ( ( ph /\ ps ) -> A. x ( ph /\ ps ) )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710

This theorem is referenced by:  bnj982  30849  bnj1351  30897  bnj1352  30898  bnj1441  30911  dvelimf-o  34214  ax12indalem  34230  ax12inda2ALT  34231  hbimpg  38770  hbimpgVD  39140