Theorem exim 1647

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 1588	. 2 |- ( A. x ( ph -> ps ) -> A. x A. x ( ph -> ps ) )
2		hbe1 1543	. 2 |- ( E. x ps -> A. x E. x ps )
3		19.8a 1638	. . . 4 |- ( ps -> E. x ps )
4	3	imim2i 12	. . 3 |- ( ( ph -> ps ) -> ( ph -> E. x ps ) )
5	4	sps 1585	. 2 |- ( A. x ( ph -> ps ) -> ( ph -> E. x ps ) )
6	1, 2, 5	exlimdh 1644	1 |- ( A. x ( ph -> ps ) -> ( E. x ph -> E. x ps ) )

Syntax hints:   -> wi 4   A.wal 1395   E.wex 1540

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 1495  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-4 1558  ax-ial 1582

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  eximi  1648  exbi  1652  eximdh  1659  19.29  1668  19.25  1674  alexim  1693  19.23t  1725  spimt  1784  equvini  1806  nfexd  1809  ax10oe  1845  sbcof2  1858  spsbim  1891  nf5-1  2077  mor  2122  rexim  2626  elex22  2818  elex2  2819  vtoclegft  2878  spcimgft  2882  spcimegft  2884  spc2gv  2897  spc3gv  2899  ssoprab2  6076  bj-inf2vnlem1  16565