Theorem exim 1652

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 1593	. 2 |- ( A. x ( ph -> ps ) -> A. x A. x ( ph -> ps ) )
2		hbe1 1548	. 2 |- ( E. x ps -> A. x E. x ps )
3		19.8a 1643	. . . 4 |- ( ps -> E. x ps )
4	3	imim2i 12	. . 3 |- ( ( ph -> ps ) -> ( ph -> E. x ps ) )
5	4	sps 1590	. 2 |- ( A. x ( ph -> ps ) -> ( ph -> E. x ps ) )
6	1, 2, 5	exlimdh 1649	1 |- ( A. x ( ph -> ps ) -> ( E. x ph -> E. x ps ) )

Syntax hints:   -> wi 4   A.wal 1400   E.wex 1545

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 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-ial 1587

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  eximi  1653  exbi  1657  eximdh  1664  19.29  1673  19.25  1679  alexim  1698  19.23t  1729  spimt  1789  equvini  1811  nfexd  1814  ax10oe  1850  sbcof2  1863  spsbim  1896  nf5-1  2084  mor  2129  rexim  2644  elex22  2837  elex2  2838  vtoclegft  2897  spcimgft  2901  spcimegft  2903  spc2gv  2916  spc3gv  2918  ssoprab2  6134  bj-inf2vnlem1  16910