Theorem exlimih 1570

Description: Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.)

Hypotheses

Ref	Expression
exlimih.1	|- ( ps -> A. x ps )
exlimih.2	|- ( ph -> ps )

Assertion

Ref	Expression
exlimih	|- ( E. x ph -> ps )

Proof of Theorem exlimih

Step	Hyp	Ref	Expression
1		exlimih.1	. . 3 |- ( ps -> A. x ps )
2	1	19.23h 1475	. 2 |- ( A. x ( ph -> ps ) <-> ( E. x ph -> ps ) )
3		exlimih.2	. 2 |- ( ph -> ps )
4	2, 3	mpgbi 1429	1 |- ( E. x ph -> ps )

Syntax hints:   -> wi 4   A.wal 1330   E.wex 1469

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-gen 1426  ax-ie2 1471

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  exlimi  1571  exlimiv  1575  19.43  1605  hbex  1613  ax6blem  1627  19.41h  1662  ax9o  1675  equid  1678  equsex  1705  cbvexh  1732  equs5a  1771  sb5rf  1829  equvin  1840  euan  2059  moexexdc  2087