Theorem exim 1645

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 1586	. 2 |- ( A. x ( ph -> ps ) -> A. x A. x ( ph -> ps ) )
2		hbe1 1541	. 2 |- ( E. x ps -> A. x E. x ps )
3		19.8a 1636	. . . 4 |- ( ps -> E. x ps )
4	3	imim2i 12	. . 3 |- ( ( ph -> ps ) -> ( ph -> E. x ps ) )
5	4	sps 1583	. 2 |- ( A. x ( ph -> ps ) -> ( ph -> E. x ps ) )
6	1, 2, 5	exlimdh 1642	1 |- ( A. x ( ph -> ps ) -> ( E. x ph -> E. x ps ) )

Syntax hints:   -> wi 4   A.wal 1393   E.wex 1538

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 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-ial 1580

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  eximi  1646  exbi  1650  eximdh  1657  19.29  1666  19.25  1672  alexim  1691  19.23t  1723  spimt  1782  equvini  1804  nfexd  1807  ax10oe  1843  sbcof2  1856  spsbim  1889  nf5-1  2075  mor  2120  rexim  2624  elex22  2815  elex2  2816  vtoclegft  2875  spcimgft  2879  spcimegft  2881  spc2gv  2894  spc3gv  2896  ssoprab2  6060  bj-inf2vnlem1  16333