Theorem exim 1592

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 1533	. 2 |- ( A. x ( ph -> ps ) -> A. x A. x ( ph -> ps ) )
2		hbe1 1488	. 2 |- ( E. x ps -> A. x E. x ps )
3		19.8a 1583	. . . 4 |- ( ps -> E. x ps )
4	3	imim2i 12	. . 3 |- ( ( ph -> ps ) -> ( ph -> E. x ps ) )
5	4	sps 1530	. 2 |- ( A. x ( ph -> ps ) -> ( ph -> E. x ps ) )
6	1, 2, 5	exlimdh 1589	1 |- ( A. x ( ph -> ps ) -> ( E. x ph -> E. x ps ) )

Syntax hints:   -> wi 4   A.wal 1346   E.wex 1485

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 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-ial 1527

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  eximi  1593  exbi  1597  eximdh  1604  19.29  1613  19.25  1619  alexim  1638  19.23t  1670  spimt  1729  equvini  1751  nfexd  1754  ax10oe  1790  sbcof2  1803  spsbim  1836  nf5-1  2017  mor  2061  rexim  2564  elex22  2745  elex2  2746  vtoclegft  2802  spcimgft  2806  spcimegft  2808  spc2gv  2821  spc3gv  2823  ssoprab2  5909  bj-inf2vnlem1  14005