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Theorem 19.23h 1432
Description: Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 1-Feb-2015.)
Hypothesis
Ref Expression
19.23h.1  |-  ( ps 
->  A. x ps )
Assertion
Ref Expression
19.23h  |-  ( A. x ( ph  ->  ps )  <->  ( E. x ph  ->  ps ) )

Proof of Theorem 19.23h
StepHypRef Expression
1 19.23h.1 . . 3  |-  ( ps 
->  A. x ps )
21ax-gen 1383 . 2  |-  A. x
( ps  ->  A. x ps )
3 19.23ht 1431 . 2  |-  ( A. x ( ps  ->  A. x ps )  -> 
( A. x (
ph  ->  ps )  <->  ( E. x ph  ->  ps )
) )
42, 3ax-mp 7 1  |-  ( A. x ( ph  ->  ps )  <->  ( E. x ph  ->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1287   E.wex 1426
This theorem was proved from axioms:  ax-mp 7  ax-gen 1383  ax-ie2 1428
This theorem is referenced by:  alnex  1433  19.8a  1527  exlimih  1529  exlimdh  1532  nf2  1603  equs5or  1758  19.23v  1811  pm11.53  1823
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