Theorem 19.38 1728

Description: Theorem 19.38 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Assertion

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

Proof of Theorem 19.38

Step	Hyp	Ref	Expression
1		hbe1 1548	. . 3 |- ( E. x ph -> A. x E. x ph )
2		hba1 1593	. . 3 |- ( A. x ps -> A. x A. x ps )
3	1, 2	hbim 1598	. 2 |- ( ( E. x ph -> A. x ps ) -> A. x ( E. x ph -> A. x ps ) )
4		19.8a 1643	. . 3 |- ( ph -> E. x ph )
5		ax-4 1563	. . 3 |- ( A. x ps -> ps )
6	4, 5	imim12i 59	. 2 |- ( ( E. x ph -> A. x ps ) -> ( ph -> ps ) )
7	3, 6	alrimih 1522	1 |- ( ( E. x ph -> A. x ps ) -> A. x ( ph -> ps ) )

Syntax hints:   -> wi 4   A.wal 1400   E.wex 1545

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 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-ial 1587  ax-i5r 1588

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  19.23t  1729  sbi2v  1947  mo3h  2140  ralm  3628