Theorem 19.41 1616

Description: Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 12-Jan-2018.)

Hypothesis

Ref	Expression
19.41.1	|- F/ x ps

Assertion

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

Proof of Theorem 19.41

Step	Hyp	Ref	Expression
1		19.40 1562	. . 3 |- ( E. x ( ph /\ ps ) -> ( E. x ph /\ E. x ps ) )
2		19.41.1	. . . . 5 |- F/ x ps
3	2	19.9 1575	. . . 4 |- ( E. x ps <-> ps )
4	3	anbi2i 444	. . 3 |- ( ( E. x ph /\ E. x ps ) <-> ( E. x ph /\ ps ) )
5	1, 4	sylib 120	. 2 |- ( E. x ( ph /\ ps ) -> ( E. x ph /\ ps ) )
6		pm3.21 260	. . . 4 |- ( ps -> ( ph -> ( ph /\ ps ) ) )
7	2, 6	eximd 1543	. . 3 |- ( ps -> ( E. x ph -> E. x ( ph /\ ps ) ) )
8	7	impcom 123	. 2 |- ( ( E. x ph /\ ps ) -> E. x ( ph /\ ps ) )
9	5, 8	impbii 124	1 |- ( E. x ( ph /\ ps ) <-> ( E. x ph /\ ps ) )

Syntax hints:   /\ wa 102   <-> wb 103   F/wnf 1389   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  df-nf 1390

This theorem is referenced by:  19.42  1618  eean  1847  r19.41  2509  eliunxp  4493  dfopab2  5835  dfoprab3s  5836  xpcomco  6323