Theorem exim 1599

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 1540	. 2 |- ( A. x ( ph -> ps ) -> A. x A. x ( ph -> ps ) )
2		hbe1 1495	. 2 |- ( E. x ps -> A. x E. x ps )
3		19.8a 1590	. . . 4 |- ( ps -> E. x ps )
4	3	imim2i 12	. . 3 |- ( ( ph -> ps ) -> ( ph -> E. x ps ) )
5	4	sps 1537	. 2 |- ( A. x ( ph -> ps ) -> ( ph -> E. x ps ) )
6	1, 2, 5	exlimdh 1596	1 |- ( A. x ( ph -> ps ) -> ( E. x ph -> E. x ps ) )

Syntax hints:   -> wi 4   A.wal 1351   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

This theorem is referenced by:  eximi  1600  exbi  1604  eximdh  1611  19.29  1620  19.25  1626  alexim  1645  19.23t  1677  spimt  1736  equvini  1758  nfexd  1761  ax10oe  1797  sbcof2  1810  spsbim  1843  nf5-1  2024  mor  2068  rexim  2571  elex22  2753  elex2  2754  vtoclegft  2810  spcimgft  2814  spcimegft  2816  spc2gv  2829  spc3gv  2831  ssoprab2  5931  bj-inf2vnlem1  14725