Theorem exim 1587

Description: Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Jul-2014.)

Assertion

Ref	Expression
exim	|- ( A. x ( ph -> ps ) -> ( E. x ph -> E. x ps ) )

Proof of Theorem exim

Step	Hyp	Ref	Expression
1		hba1 1528	. 2 |- ( A. x ( ph -> ps ) -> A. x A. x ( ph -> ps ) )
2		hbe1 1483	. 2 |- ( E. x ps -> A. x E. x ps )
3		19.8a 1578	. . . 4 |- ( ps -> E. x ps )
4	3	imim2i 12	. . 3 |- ( ( ph -> ps ) -> ( ph -> E. x ps ) )
5	4	sps 1525	. 2 |- ( A. x ( ph -> ps ) -> ( ph -> E. x ps ) )
6	1, 2, 5	exlimdh 1584	1 |- ( A. x ( ph -> ps ) -> ( E. x ph -> E. x ps ) )

Syntax hints:   -> wi 4   A.wal 1341   E.wex 1480

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 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-ial 1522

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  eximi  1588  exbi  1592  eximdh  1599  19.29  1608  19.25  1614  alexim  1633  19.23t  1665  spimt  1724  equvini  1746  nfexd  1749  ax10oe  1785  sbcof2  1798  spsbim  1831  nf5-1  2012  mor  2056  rexim  2560  elex22  2741  elex2  2742  vtoclegft  2798  spcimgft  2802  spcimegft  2804  spc2gv  2817  spc3gv  2819  ssoprab2  5898  bj-inf2vnlem1  13852