Theorem hban 1557

Description: If x is not free in ph and ps, it is not free in ( ph /\ ps ). (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 2-Feb-2015.)

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	. . 3 |- ( ph -> A. x ph )
2		hb.2	. . 3 |- ( ps -> A. x ps )
3	1, 2	anim12i 338	. 2 |- ( ( ph /\ ps ) -> ( A. x ph /\ A. x ps ) )
4		19.26 1491	. 2 |- ( A. x ( ph /\ ps ) <-> ( A. x ph /\ A. x ps ) )
5	3, 4	sylibr 134	1 |- ( ( ph /\ ps ) -> A. x ( ph /\ ps ) )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1361

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1457  ax-gen 1459

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  hbbi  1558  hb3an  1560  hbsbv  1952  mopick  2115  eupicka  2117  mopick2  2120  cleqh  2288