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Theorem r19.23t 2733
Description: Closed theorem form of r19.23 2734. (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 1801 . 2  |-  ( F/ x ps  ->  ( A. x ( ( x  e.  A  /\  ph )  ->  ps )  <->  ( E. x ( x  e.  A  /\  ph )  ->  ps ) ) )
2 df-ral 2624 . . 3  |-  ( A. x  e.  A  ( ph  ->  ps )  <->  A. x
( x  e.  A  ->  ( ph  ->  ps ) ) )
3 impexp 433 . . . 4  |-  ( ( ( x  e.  A  /\  ph )  ->  ps ) 
<->  ( x  e.  A  ->  ( ph  ->  ps ) ) )
43albii 1566 . . 3  |-  ( A. x ( ( x  e.  A  /\  ph )  ->  ps )  <->  A. x
( x  e.  A  ->  ( ph  ->  ps ) ) )
52, 4bitr4i 243 . 2  |-  ( A. x  e.  A  ( ph  ->  ps )  <->  A. x
( ( x  e.  A  /\  ph )  ->  ps ) )
6 df-rex 2625 . . 3  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
76imbi1i 315 . 2  |-  ( ( E. x  e.  A  ph 
->  ps )  <->  ( E. x ( x  e.  A  /\  ph )  ->  ps ) )
81, 5, 73bitr4g 279 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    <-> wb 176    /\ wa 358   A.wal 1540   E.wex 1541   F/wnf 1544    e. wcel 1710   A.wral 2619   E.wrex 2620
This theorem is referenced by:  r19.23  2734  rexlimd2  2741  riotasv3d  6440  riotasv3dOLD  6441
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-ral 2624  df-rex 2625
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