Theorem 19.29 1551

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		pm3.2 137	. . . 4 |- ( ph -> ( ps -> ( ph /\ ps ) ) )
2	1	alimi 1384	. . 3 |- ( A. x ph -> A. x ( ps -> ( ph /\ ps ) ) )
3		exim 1530	. . 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 122	1 |- ( ( A. x ph /\ E. x ps ) -> E. x ( ph /\ ps ) )

Syntax hints:   -> wi 4   /\ wa 102   A.wal 1282   E.wex 1421

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-ial 1467

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  19.29r  1552  19.29x  1554  19.35-1  1555  equs4  1653  equvini  1681  rexxfrd  4213  funimaexglem  5002  bj-inex  10698