Theorem hbim 1569

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 O'Cat, 3-Mar-2008.) (Revised by Mario Carneiro, 2-Feb-2015.)

Hypotheses

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

Assertion

Ref	Expression
hbim	|- ( ( ph -> ps ) -> A. x ( ph -> ps ) )

Proof of Theorem hbim

Step	Hyp	Ref	Expression
1		ax-4 1534	. . 3 |- ( A. x ph -> ph )
2		hb.2	. . 3 |- ( ps -> A. x ps )
3	1, 2	imim12i 59	. 2 |- ( ( ph -> ps ) -> ( A. x ph -> A. x ps ) )
4		ax-i5r 1559	. 2 |- ( ( A. x ph -> A. x ps ) -> A. x ( A. x ph -> ps ) )
5		hb.1	. . . 4 |- ( ph -> A. x ph )
6	5	imim1i 60	. . 3 |- ( ( A. x ph -> ps ) -> ( ph -> ps ) )
7	6	alimi 1479	. 2 |- ( A. x ( A. x ph -> ps ) -> A. x ( ph -> ps ) )
8	3, 4, 7	3syl 17	1 |- ( ( ph -> ps ) -> A. x ( ph -> ps ) )

Syntax hints:   -> wi 4   A.wal 1371

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-5 1471  ax-gen 1473  ax-4 1534  ax-i5r 1559

This theorem is referenced by:  hbbi  1572  hbia1  1576  19.21h  1581  19.38  1700  hbsbv  1970  hbmo1  2093  hbmo  2094  moexexdc  2140  2eu4  2149  cleqh  2307  hbral  2537