Theorem 19.29r 1581

Description: Variation of Theorem 19.29 of [Margaris] p. 90. (Contributed by NM, 18-Aug-1993.)

Assertion

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

Proof of Theorem 19.29r

Step	Hyp	Ref	Expression
1		19.29 1580	. 2 |- ( ( A. x ps /\ E. x ph ) -> E. x ( ps /\ ph ) )
2		ancom 264	. 2 |- ( ( E. x ph /\ A. x ps ) <-> ( A. x ps /\ E. x ph ) )
3		exancom 1568	. 2 |- ( E. x ( ph /\ ps ) <-> E. x ( ps /\ ph ) )
4	1, 2, 3	3imtr4i 200	1 |- ( ( E. x ph /\ A. x ps ) -> E. x ( ph /\ ps ) )

Syntax hints:   -> wi 4   /\ wa 103   A.wal 1310   E.wex 1449

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 1404  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-4 1468  ax-ial 1495

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  19.29r2  1582  19.29x  1583  exan  1652  ax9o  1657  equvini  1712  eu2  2017  intab  3764  imadiflem  5158  bj-inex  12788