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Theorem 19.21vOLDOLD 1789
Description: Obsolete proof of 19.21v 1788 as of 12-Jul-2020. (Contributed by NM, 21-Jun-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 2-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
19.21vOLDOLD  |-  ( A. x ( ph  ->  ps )  <->  ( ph  ->  A. x ps ) )
Distinct variable group:    ph, x
Allowed substitution hint:    ps( x)

Proof of Theorem 19.21vOLDOLD
StepHypRef Expression
1 ax-5 1760 . . 3  |-  ( ph  ->  A. x ph )
2 alim 1685 . . 3  |-  ( A. x ( ph  ->  ps )  ->  ( A. x ph  ->  A. x ps ) )
31, 2syl5 33 . 2  |-  ( A. x ( ph  ->  ps )  ->  ( ph  ->  A. x ps )
)
4 ax5e 1762 . . . 4  |-  ( E. x ph  ->  ph )
54imim1i 60 . . 3  |-  ( (
ph  ->  A. x ps )  ->  ( E. x ph  ->  A. x ps )
)
6 19.38 1714 . . 3  |-  ( ( E. x ph  ->  A. x ps )  ->  A. x ( ph  ->  ps ) )
75, 6syl 17 . 2  |-  ( (
ph  ->  A. x ps )  ->  A. x ( ph  ->  ps ) )
83, 7impbii 191 1  |-  ( A. x ( ph  ->  ps )  <->  ( ph  ->  A. x ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188   A.wal 1444   E.wex 1665
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760
This theorem depends on definitions:  df-bi 189  df-ex 1666
This theorem is referenced by: (None)
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