MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  19.23tOLD Unicode version

Theorem 19.23tOLD 1838
Description: Obsolete proof of 19.23t 1818 as of 1-Jan-2018. (Contributed by NM, 7-Nov-2005.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
19.23tOLD  |-  ( F/ x ps  ->  ( A. x ( ph  ->  ps )  <->  ( E. x ph  ->  ps ) ) )

Proof of Theorem 19.23tOLD
StepHypRef Expression
1 exim 1584 . . 3  |-  ( A. x ( ph  ->  ps )  ->  ( E. x ph  ->  E. x ps ) )
2 19.9t 1793 . . . 4  |-  ( F/ x ps  ->  ( E. x ps  <->  ps )
)
32imbi2d 308 . . 3  |-  ( F/ x ps  ->  (
( E. x ph  ->  E. x ps )  <->  ( E. x ph  ->  ps ) ) )
41, 3syl5ib 211 . 2  |-  ( F/ x ps  ->  ( A. x ( ph  ->  ps )  ->  ( E. x ph  ->  ps )
) )
5 nfnf1 1808 . . 3  |-  F/ x F/ x ps
6 nfe1 1747 . . . . 5  |-  F/ x E. x ph
76a1i 11 . . . 4  |-  ( F/ x ps  ->  F/ x E. x ph )
8 id 20 . . . 4  |-  ( F/ x ps  ->  F/ x ps )
97, 8nfimd 1827 . . 3  |-  ( F/ x ps  ->  F/ x ( E. x ph  ->  ps ) )
10 19.8a 1762 . . . . 5  |-  ( ph  ->  E. x ph )
1110a1i 11 . . . 4  |-  ( F/ x ps  ->  ( ph  ->  E. x ph )
)
1211imim1d 71 . . 3  |-  ( F/ x ps  ->  (
( E. x ph  ->  ps )  ->  ( ph  ->  ps ) ) )
135, 9, 12alrimdd 1784 . 2  |-  ( F/ x ps  ->  (
( E. x ph  ->  ps )  ->  A. x
( ph  ->  ps )
) )
144, 13impbid 184 1  |-  ( F/ x ps  ->  ( A. x ( ph  ->  ps )  <->  ( E. x ph  ->  ps ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1549   E.wex 1550   F/wnf 1553
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-11 1761
This theorem depends on definitions:  df-bi 178  df-ex 1551  df-nf 1554
  Copyright terms: Public domain W3C validator