Theorem 19.21h 1555

Description: Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as " x is not free in ph". New proofs should use 19.21 1581 instead. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Hypothesis

Ref	Expression
19.21h.1	|- ( ph -> A. x ph )

Assertion

Ref	Expression
19.21h	|- ( A. x ( ph -> ps ) <-> ( ph -> A. x ps ) )

Proof of Theorem 19.21h

Step	Hyp	Ref	Expression
1		19.21h.1	. . 3 |- ( ph -> A. x ph )
2		alim 1455	. . 3 |- ( A. x ( ph -> ps ) -> ( A. x ph -> A. x ps ) )
3	1, 2	syl5 32	. 2 |- ( A. x ( ph -> ps ) -> ( ph -> A. x ps ) )
4		hba1 1538	. . . 4 |- ( A. x ps -> A. x A. x ps )
5	1, 4	hbim 1543	. . 3 |- ( ( ph -> A. x ps ) -> A. x ( ph -> A. x ps ) )
6		ax-4 1508	. . . 4 |- ( A. x ps -> ps )
7	6	imim2i 12	. . 3 |- ( ( ph -> A. x ps ) -> ( ph -> ps ) )
8	5, 7	alrimih 1467	. 2 |- ( ( ph -> A. x ps ) -> A. x ( ph -> ps ) )
9	3, 8	impbii 126	1 |- ( A. x ( ph -> ps ) <-> ( ph -> A. x ps ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia2 107  ax-ia3 108  ax-5 1445  ax-gen 1447  ax-4 1508  ax-ial 1532  ax-i5r 1533

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  hbim1  1568  nf3  1667  19.21v  1871