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Theorem r19.23t 2440
Description: Closed theorem form of r19.23 2441. (Contributed by NM, 4-Mar-2013.) (Revised by Mario Carneiro, 8-Oct-2016.)
Assertion
Ref Expression
r19.23t  |-  ( F/ x ps  ->  ( A. x  e.  A  ( ph  ->  ps )  <->  ( E. x  e.  A  ph 
->  ps ) ) )

Proof of Theorem r19.23t
StepHypRef Expression
1 19.23t 1583 . 2  |-  ( F/ x ps  ->  ( A. x ( ( x  e.  A  /\  ph )  ->  ps )  <->  ( E. x ( x  e.  A  /\  ph )  ->  ps ) ) )
2 df-ral 2328 . . 3  |-  ( A. x  e.  A  ( ph  ->  ps )  <->  A. x
( x  e.  A  ->  ( ph  ->  ps ) ) )
3 impexp 254 . . . 4  |-  ( ( ( x  e.  A  /\  ph )  ->  ps ) 
<->  ( x  e.  A  ->  ( ph  ->  ps ) ) )
43albii 1375 . . 3  |-  ( A. x ( ( x  e.  A  /\  ph )  ->  ps )  <->  A. x
( x  e.  A  ->  ( ph  ->  ps ) ) )
52, 4bitr4i 180 . 2  |-  ( A. x  e.  A  ( ph  ->  ps )  <->  A. x
( ( x  e.  A  /\  ph )  ->  ps ) )
6 df-rex 2329 . . 3  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
76imbi1i 231 . 2  |-  ( ( E. x  e.  A  ph 
->  ps )  <->  ( E. x ( x  e.  A  /\  ph )  ->  ps ) )
81, 5, 73bitr4g 216 1  |-  ( F/ x ps  ->  ( A. x  e.  A  ( ph  ->  ps )  <->  ( E. x  e.  A  ph 
->  ps ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102   A.wal 1257   F/wnf 1365   E.wex 1397    e. wcel 1409   A.wral 2323   E.wrex 2324
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-ial 1443  ax-i5r 1444
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-ral 2328  df-rex 2329
This theorem is referenced by:  r19.23  2441  rexlimd2  2448
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