Theorem exbi 1591

Description: Theorem 19.18 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Assertion

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

Proof of Theorem exbi

Step	Hyp	Ref	Expression
1		biimp 117	. . . 4 |- ( ( ph <-> ps ) -> ( ph -> ps ) )
2	1	alimi 1442	. . 3 |- ( A. x ( ph <-> ps ) -> A. x ( ph -> ps ) )
3		exim 1586	. . 3 |- ( A. x ( ph -> ps ) -> ( E. x ph -> E. x ps ) )
4	2, 3	syl 14	. 2 |- ( A. x ( ph <-> ps ) -> ( E. x ph -> E. x ps ) )
5		biimpr 129	. . . 4 |- ( ( ph <-> ps ) -> ( ps -> ph ) )
6	5	alimi 1442	. . 3 |- ( A. x ( ph <-> ps ) -> A. x ( ps -> ph ) )
7		exim 1586	. . 3 |- ( A. x ( ps -> ph ) -> ( E. x ps -> E. x ph ) )
8	6, 7	syl 14	. 2 |- ( A. x ( ph <-> ps ) -> ( E. x ps -> E. x ph ) )
9	4, 8	impbid 128	1 |- ( A. x ( ph <-> ps ) -> ( E. x ph <-> E. x ps ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1340   E.wex 1479

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1434  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-4 1497  ax-ial 1521

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  exbii  1592  exbidh  1601  exintrbi  1620  19.19  1653  rexrnmpt  5622