Theorem 19.29 1567

Description: Theorem 19.29 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.)

Assertion

Ref	Expression
19.29	|- ( ( A. x ph /\ E. x ps ) -> E. x ( ph /\ ps ) )

Proof of Theorem 19.29

Step	Hyp	Ref	Expression
1		pm3.2 138	. . . 4 |- ( ph -> ( ps -> ( ph /\ ps ) ) )
2	1	alimi 1399	. . 3 |- ( A. x ph -> A. x ( ps -> ( ph /\ ps ) ) )
3		exim 1546	. . 3 |- ( A. x ( ps -> ( ph /\ ps ) ) -> ( E. x ps -> E. x ( ph /\ ps ) ) )
4	2, 3	syl 14	. 2 |- ( A. x ph -> ( E. x ps -> E. x ( ph /\ ps ) ) )
5	4	imp 123	1 |- ( ( A. x ph /\ E. x ps ) -> E. x ( ph /\ ps ) )

Syntax hints:   -> wi 4   /\ wa 103   A.wal 1297   E.wex 1436

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1391  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-4 1455  ax-ial 1482

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  19.29r  1568  19.29x  1570  19.35-1  1571  equs4  1671  equvini  1699  rexxfrd  4322  funimaexglem  5142  bj-inex  12686