Theorem exim 1622

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 1563	. 2 |- ( A. x ( ph -> ps ) -> A. x A. x ( ph -> ps ) )
2		hbe1 1518	. 2 |- ( E. x ps -> A. x E. x ps )
3		19.8a 1613	. . . 4 |- ( ps -> E. x ps )
4	3	imim2i 12	. . 3 |- ( ( ph -> ps ) -> ( ph -> E. x ps ) )
5	4	sps 1560	. 2 |- ( A. x ( ph -> ps ) -> ( ph -> E. x ps ) )
6	1, 2, 5	exlimdh 1619	1 |- ( A. x ( ph -> ps ) -> ( E. x ph -> E. x ps ) )

Syntax hints:   -> wi 4   A.wal 1371   E.wex 1515

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 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-4 1533  ax-ial 1557

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  eximi  1623  exbi  1627  eximdh  1634  19.29  1643  19.25  1649  alexim  1668  19.23t  1700  spimt  1759  equvini  1781  nfexd  1784  ax10oe  1820  sbcof2  1833  spsbim  1866  nf5-1  2052  mor  2096  rexim  2600  elex22  2787  elex2  2788  vtoclegft  2845  spcimgft  2849  spcimegft  2851  spc2gv  2864  spc3gv  2866  ssoprab2  6001  bj-inf2vnlem1  15906