Theorem 19.42 2105

Description: Theorem 19.42 of [Margaris] p. 90. See 19.42v 1918 for a version requiring fewer axioms. See exan 1788 for an immediate version. (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 2103	. 2 |- ( E. x ( ps /\ ph ) <-> ( E. x ps /\ ph ) )
3		exancom 1787	. 2 |- ( E. x ( ph /\ ps ) <-> E. x ( ps /\ ph ) )
4		ancom 466	. 2 |- ( ( ph /\ E. x ps ) <-> ( E. x ps /\ ph ) )
5	2, 3, 4	3bitr4i 292	1 |- ( E. x ( ph /\ ps ) <-> ( ph /\ E. x ps ) )

Syntax hints:   <-> wb 196   /\ wa 384   E.wex 1704   F/wnf 1708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-nf 1710

This theorem is referenced by:  eean  2181  bnj916  31003  bnj983  31021