Theorem exim 1613

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 1554	. 2 |- ( A. x ( ph -> ps ) -> A. x A. x ( ph -> ps ) )
2		hbe1 1509	. 2 |- ( E. x ps -> A. x E. x ps )
3		19.8a 1604	. . . 4 |- ( ps -> E. x ps )
4	3	imim2i 12	. . 3 |- ( ( ph -> ps ) -> ( ph -> E. x ps ) )
5	4	sps 1551	. 2 |- ( A. x ( ph -> ps ) -> ( ph -> E. x ps ) )
6	1, 2, 5	exlimdh 1610	1 |- ( A. x ( ph -> ps ) -> ( E. x ph -> E. x ps ) )

Syntax hints:   -> wi 4   A.wal 1362   E.wex 1506

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-ial 1548

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  eximi  1614  exbi  1618  eximdh  1625  19.29  1634  19.25  1640  alexim  1659  19.23t  1691  spimt  1750  equvini  1772  nfexd  1775  ax10oe  1811  sbcof2  1824  spsbim  1857  nf5-1  2043  mor  2087  rexim  2591  elex22  2778  elex2  2779  vtoclegft  2836  spcimgft  2840  spcimegft  2842  spc2gv  2855  spc3gv  2857  ssoprab2  5982  bj-inf2vnlem1  15700