Theorem 19.41 2028

Description: Theorem 19.41 of [Margaris] p. 90. See 19.41v 1823 for a version requiring fewer axioms. (Contributed by NM, 14-May-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 1725	. . 3 |- ( E. x ( ph /\ ps ) -> ( E. x ph /\ E. x ps ) )
2		19.41.1	. . . . 5 |- F/ x ps
3	2	19.9 1947	. . . 4 |- ( E. x ps <-> ps )
4	3	anbi2i 698	. . 3 |- ( ( E. x ph /\ E. x ps ) <-> ( E. x ph /\ ps ) )
5	1, 4	sylib 199	. 2 |- ( E. x ( ph /\ ps ) -> ( E. x ph /\ ps ) )
6		pm3.21 449	. . . 4 |- ( ps -> ( ph -> ( ph /\ ps ) ) )
7	2, 6	eximd 1937	. . 3 |- ( ps -> ( E. x ph -> E. x ( ph /\ ps ) ) )
8	7	impcom 431	. 2 |- ( ( E. x ph /\ ps ) -> E. x ( ph /\ ps ) )
9	5, 8	impbii 190	1 |- ( E. x ( ph /\ ps ) <-> ( E. x ph /\ ps ) )

Syntax hints:   <-> wb 187   /\ wa 370   E.wex 1657   F/wnf 1661

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-12 1909

This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1658  df-nf 1662

This theorem is referenced by:  19.42  2029  exan  2030  eean  2052  eeeanv  2054  equsexALT  2103  2sb5rf  2257  r19.41  2920  eliunxp  4934  dfopab2  6805  dfoprab3s  6806  xpcomco  7615  mpt2mptxf  28226  bnj605  29670  bnj607  29679  bj-equsexv  31252  2sb5nd  36840  2sb5ndVD  37223  2sb5ndALT  37245  eliunxp2  39718