Theorem 19.42 1681

Description: Theorem 19.42 of [Margaris] p. 90. (Contributed by NM, 18-Aug-1993.)

Hypothesis

Ref	Expression
19.42.1	|- F/ x ph

Assertion

Ref	Expression
19.42	|- ( E. x ( ph /\ ps ) <-> ( ph /\ E. x ps ) )

Proof of Theorem 19.42

Step	Hyp	Ref	Expression
1		19.42.1	. . 3 |- F/ x ph
2	1	19.41 1679	. 2 |- ( E. x ( ps /\ ph ) <-> ( E. x ps /\ ph ) )
3		exancom 1601	. 2 |- ( E. x ( ph /\ ps ) <-> E. x ( ps /\ ph ) )
4		ancom 264	. 2 |- ( ( ph /\ E. x ps ) <-> ( E. x ps /\ ph ) )
5	2, 3, 4	3bitr4i 211	1 |- ( E. x ( ph /\ ps ) <-> ( ph /\ E. x ps ) )

Syntax hints:   /\ wa 103   <-> wb 104   F/wnf 1453   E.wex 1485

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 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-nf 1454

This theorem is referenced by:  eean  1924  r2exf  2488