Theorem 19.41 1686

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 1631	. . 3 |- ( E. x ( ph /\ ps ) -> ( E. x ph /\ E. x ps ) )
2		19.41.1	. . . . 5 |- F/ x ps
3	2	19.9 1644	. . . 4 |- ( E. x ps <-> ps )
4	3	anbi2i 457	. . 3 |- ( ( E. x ph /\ E. x ps ) <-> ( E. x ph /\ ps ) )
5	1, 4	sylib 122	. 2 |- ( E. x ( ph /\ ps ) -> ( E. x ph /\ ps ) )
6		pm3.21 264	. . . 4 |- ( ps -> ( ph -> ( ph /\ ps ) ) )
7	2, 6	eximd 1612	. . 3 |- ( ps -> ( E. x ph -> E. x ( ph /\ ps ) ) )
8	7	impcom 125	. 2 |- ( ( E. x ph /\ ps ) -> E. x ( ph /\ ps ) )
9	5, 8	impbii 126	1 |- ( E. x ( ph /\ ps ) <-> ( E. x ph /\ ps ) )

Syntax hints:   /\ wa 104   <-> wb 105   F/wnf 1460   E.wex 1492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-ial 1534

This theorem depends on definitions:  df-bi 117  df-nf 1461

This theorem is referenced by:  19.42  1688  eean  1931  r19.41  2632  eliunxp  4765  dfopab2  6187  dfoprab3s  6188  xpcomco  6823