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Theorem rexbidar 31559
Description: More general form of rexbida 2901. (Contributed by Andrew Salmon, 25-Jul-2011.)
Hypotheses
Ref Expression
ralbidar.1  |-  ( ph  ->  A. x  e.  A  ph )
ralbidar.2  |-  ( (
ph  /\  x  e.  A )  ->  ( ps 
<->  ch ) )
Assertion
Ref Expression
rexbidar  |-  ( ph  ->  ( E. x  e.  A  ps  <->  E. x  e.  A  ch )
)

Proof of Theorem rexbidar
StepHypRef Expression
1 ralbidar.1 . . . . 5  |-  ( ph  ->  A. x  e.  A  ph )
2 ralbidar.2 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  ( ps 
<->  ch ) )
32ex 432 . . . . . 6  |-  ( ph  ->  ( x  e.  A  ->  ( ps  <->  ch )
) )
43ralimi 2785 . . . . 5  |-  ( A. x  e.  A  ph  ->  A. x  e.  A  ( x  e.  A  -> 
( ps  <->  ch )
) )
51, 4syl 16 . . . 4  |-  ( ph  ->  A. x  e.  A  ( x  e.  A  ->  ( ps  <->  ch )
) )
6 df-ral 2747 . . . 4  |-  ( A. x  e.  A  (
x  e.  A  -> 
( ps  <->  ch )
)  <->  A. x ( x  e.  A  ->  (
x  e.  A  -> 
( ps  <->  ch )
) ) )
75, 6sylib 196 . . 3  |-  ( ph  ->  A. x ( x  e.  A  ->  (
x  e.  A  -> 
( ps  <->  ch )
) ) )
8 pm2.43 51 . . . . 5  |-  ( ( x  e.  A  -> 
( x  e.  A  ->  ( ps  <->  ch )
) )  ->  (
x  e.  A  -> 
( ps  <->  ch )
) )
98pm5.32d 637 . . . 4  |-  ( ( x  e.  A  -> 
( x  e.  A  ->  ( ps  <->  ch )
) )  ->  (
( x  e.  A  /\  ps )  <->  ( x  e.  A  /\  ch )
) )
109alimi 1648 . . 3  |-  ( A. x ( x  e.  A  ->  ( x  e.  A  ->  ( ps  <->  ch ) ) )  ->  A. x ( ( x  e.  A  /\  ps ) 
<->  ( x  e.  A  /\  ch ) ) )
11 exbi 1681 . . 3  |-  ( A. x ( ( x  e.  A  /\  ps ) 
<->  ( x  e.  A  /\  ch ) )  -> 
( E. x ( x  e.  A  /\  ps )  <->  E. x ( x  e.  A  /\  ch ) ) )
127, 10, 113syl 20 . 2  |-  ( ph  ->  ( E. x ( x  e.  A  /\  ps )  <->  E. x ( x  e.  A  /\  ch ) ) )
13 df-rex 2748 . 2  |-  ( E. x  e.  A  ps  <->  E. x ( x  e.  A  /\  ps )
)
14 df-rex 2748 . 2  |-  ( E. x  e.  A  ch  <->  E. x ( x  e.  A  /\  ch )
)
1512, 13, 143bitr4g 288 1  |-  ( ph  ->  ( E. x  e.  A  ps  <->  E. x  e.  A  ch )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367   A.wal 1397   E.wex 1627    e. wcel 1836   A.wral 2742   E.wrex 2743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646
This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1628  df-ral 2747  df-rex 2748
This theorem is referenced by: (None)
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