Theorem hbim 2127

Description: If x is not free in ph and ps, it is not free in ( ph -> ps ). (Contributed by NM, 24-Jan-1993.) (Proof shortened by Mel L. O'Cat, 3-Mar-2008.) (Proof shortened by Wolf Lammen, 1-Jan-2018.)

Hypotheses

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

Assertion

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

Proof of Theorem hbim

Step	Hyp	Ref	Expression
1		hbim.1	. 2 |- ( ph -> A. x ph )
2		hbim.2	. . 3 |- ( ps -> A. x ps )
3	2	a1i 11	. 2 |- ( ph -> ( ps -> A. x ps ) )
4	1, 3	hbim1 2125	1 |- ( ( ph -> ps ) -> A. x ( ph -> ps ) )

Syntax hints:   -> wi 4   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-ex 1705  df-nf 1710

This theorem is referenced by:  axi5r  2594  hbral  2943