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Theorem ceqsalt 2684
Description: Closed theorem version of ceqsalg 2686. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)
Assertion
Ref Expression
ceqsalt  |-  ( ( F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) )  /\  A  e.  V )  ->  ( A. x ( x  =  A  ->  ph )  <->  ps )
)
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    ps( x)    V( x)

Proof of Theorem ceqsalt
StepHypRef Expression
1 elisset 2672 . . . 4  |-  ( A  e.  V  ->  E. x  x  =  A )
213ad2ant3 987 . . 3  |-  ( ( F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) )  /\  A  e.  V )  ->  E. x  x  =  A )
3 bi1 117 . . . . . . 7  |-  ( (
ph 
<->  ps )  ->  ( ph  ->  ps ) )
43imim3i 61 . . . . . 6  |-  ( ( x  =  A  -> 
( ph  <->  ps ) )  -> 
( ( x  =  A  ->  ph )  -> 
( x  =  A  ->  ps ) ) )
54al2imi 1417 . . . . 5  |-  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  ->  ( A. x ( x  =  A  ->  ph )  ->  A. x ( x  =  A  ->  ps )
) )
653ad2ant2 986 . . . 4  |-  ( ( F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) )  /\  A  e.  V )  ->  ( A. x ( x  =  A  ->  ph )  ->  A. x ( x  =  A  ->  ps )
) )
7 19.23t 1638 . . . . 5  |-  ( F/ x ps  ->  ( A. x ( x  =  A  ->  ps )  <->  ( E. x  x  =  A  ->  ps )
) )
873ad2ant1 985 . . . 4  |-  ( ( F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) )  /\  A  e.  V )  ->  ( A. x ( x  =  A  ->  ps )  <->  ( E. x  x  =  A  ->  ps )
) )
96, 8sylibd 148 . . 3  |-  ( ( F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) )  /\  A  e.  V )  ->  ( A. x ( x  =  A  ->  ph )  -> 
( E. x  x  =  A  ->  ps ) ) )
102, 9mpid 42 . 2  |-  ( ( F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) )  /\  A  e.  V )  ->  ( A. x ( x  =  A  ->  ph )  ->  ps ) )
11 bi2 129 . . . . . . 7  |-  ( (
ph 
<->  ps )  ->  ( ps  ->  ph ) )
1211imim2i 12 . . . . . 6  |-  ( ( x  =  A  -> 
( ph  <->  ps ) )  -> 
( x  =  A  ->  ( ps  ->  ph ) ) )
1312com23 78 . . . . 5  |-  ( ( x  =  A  -> 
( ph  <->  ps ) )  -> 
( ps  ->  (
x  =  A  ->  ph ) ) )
1413alimi 1414 . . . 4  |-  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  ->  A. x
( ps  ->  (
x  =  A  ->  ph ) ) )
15143ad2ant2 986 . . 3  |-  ( ( F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) )  /\  A  e.  V )  ->  A. x
( ps  ->  (
x  =  A  ->  ph ) ) )
16 19.21t 1544 . . . 4  |-  ( F/ x ps  ->  ( A. x ( ps  ->  ( x  =  A  ->  ph ) )  <->  ( ps  ->  A. x ( x  =  A  ->  ph )
) ) )
17163ad2ant1 985 . . 3  |-  ( ( F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) )  /\  A  e.  V )  ->  ( A. x ( ps  ->  ( x  =  A  ->  ph ) )  <->  ( ps  ->  A. x ( x  =  A  ->  ph )
) ) )
1815, 17mpbid 146 . 2  |-  ( ( F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) )  /\  A  e.  V )  ->  ( ps  ->  A. x ( x  =  A  ->  ph )
) )
1910, 18impbid 128 1  |-  ( ( F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) )  /\  A  e.  V )  ->  ( A. x ( x  =  A  ->  ph )  <->  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    /\ w3a 945   A.wal 1312    = wceq 1314   F/wnf 1419   E.wex 1451    e. wcel 1463
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3an 947  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-v 2660
This theorem is referenced by:  ceqsralt  2685
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