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Theorem ceqsalt 3129
Description: Closed theorem version of ceqsalg 3131. (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 3117 . . . 4  |-  ( A  e.  V  ->  E. x  x  =  A )
213ad2ant3 1017 . . 3  |-  ( ( F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) )  /\  A  e.  V )  ->  E. x  x  =  A )
3 bi1 186 . . . . . . 7  |-  ( (
ph 
<->  ps )  ->  ( ph  ->  ps ) )
43imim3i 59 . . . . . 6  |-  ( ( x  =  A  -> 
( ph  <->  ps ) )  -> 
( ( x  =  A  ->  ph )  -> 
( x  =  A  ->  ps ) ) )
54al2imi 1641 . . . . 5  |-  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  ->  ( A. x ( x  =  A  ->  ph )  ->  A. x ( x  =  A  ->  ps )
) )
653ad2ant2 1016 . . . 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 1914 . . . . 5  |-  ( F/ x ps  ->  ( A. x ( x  =  A  ->  ps )  <->  ( E. x  x  =  A  ->  ps )
) )
873ad2ant1 1015 . . . 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 214 . . 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 41 . 2  |-  ( ( F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) )  /\  A  e.  V )  ->  ( A. x ( x  =  A  ->  ph )  ->  ps ) )
11 bi2 198 . . . . . . 7  |-  ( (
ph 
<->  ps )  ->  ( ps  ->  ph ) )
1211imim2i 14 . . . . . 6  |-  ( ( x  =  A  -> 
( ph  <->  ps ) )  -> 
( x  =  A  ->  ( ps  ->  ph ) ) )
1312com23 78 . . . . 5  |-  ( ( x  =  A  -> 
( ph  <->  ps ) )  -> 
( ps  ->  (
x  =  A  ->  ph ) ) )
1413alimi 1638 . . . 4  |-  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  ->  A. x
( ps  ->  (
x  =  A  ->  ph ) ) )
15143ad2ant2 1016 . . 3  |-  ( ( F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) )  /\  A  e.  V )  ->  A. x
( ps  ->  (
x  =  A  ->  ph ) ) )
16 19.21t 1909 . . . 4  |-  ( F/ x ps  ->  ( A. x ( ps  ->  ( x  =  A  ->  ph ) )  <->  ( ps  ->  A. x ( x  =  A  ->  ph )
) ) )
17163ad2ant1 1015 . . 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 210 . 2  |-  ( ( F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) )  /\  A  e.  V )  ->  ( ps  ->  A. x ( x  =  A  ->  ph )
) )
1910, 18impbid 191 1  |-  ( ( F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) )  /\  A  e.  V )  ->  ( A. x ( x  =  A  ->  ph )  <->  ps )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 971   A.wal 1396    = wceq 1398   E.wex 1617   F/wnf 1621    e. wcel 1823
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-12 1859  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-v 3108
This theorem is referenced by:  ceqsralt  3130  ceqsalg  3131
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