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Theorem vtoclgft 3008
Description: Closed theorem form of vtoclgf 3016. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 12-Oct-2016.)
Assertion
Ref Expression
vtoclgft  |-  ( ( ( F/_ x A  /\  F/ x ps )  /\  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  /\  A. x ph )  /\  A  e.  V )  ->  ps )

Proof of Theorem vtoclgft
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 elex 2970 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 elisset 2972 . . . . 5  |-  ( A  e.  _V  ->  E. z 
z  =  A )
323ad2ant3 981 . . . 4  |-  ( ( ( F/_ x A  /\  F/ x ps )  /\  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  /\  A. x ph )  /\  A  e.  _V )  ->  E. z 
z  =  A )
4 nfnfc1 2581 . . . . . . 7  |-  F/ x F/_ x A
5 nfcvd 2579 . . . . . . . 8  |-  ( F/_ x A  ->  F/_ x
z )
6 id 21 . . . . . . . 8  |-  ( F/_ x A  ->  F/_ x A )
75, 6nfeqd 2592 . . . . . . 7  |-  ( F/_ x A  ->  F/ x  z  =  A )
8 eqeq1 2448 . . . . . . . 8  |-  ( z  =  x  ->  (
z  =  A  <->  x  =  A ) )
98a1i 11 . . . . . . 7  |-  ( F/_ x A  ->  ( z  =  x  ->  (
z  =  A  <->  x  =  A ) ) )
104, 7, 9cbvexd 1991 . . . . . 6  |-  ( F/_ x A  ->  ( E. z  z  =  A  <->  E. x  x  =  A ) )
1110ad2antrr 708 . . . . 5  |-  ( ( ( F/_ x A  /\  F/ x ps )  /\  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  /\  A. x ph ) )  -> 
( E. z  z  =  A  <->  E. x  x  =  A )
)
12113adant3 978 . . . 4  |-  ( ( ( F/_ x A  /\  F/ x ps )  /\  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  /\  A. x ph )  /\  A  e.  _V )  ->  ( E. z  z  =  A 
<->  E. x  x  =  A ) )
133, 12mpbid 203 . . 3  |-  ( ( ( F/_ x A  /\  F/ x ps )  /\  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  /\  A. x ph )  /\  A  e.  _V )  ->  E. x  x  =  A )
14 bi1 180 . . . . . . . . 9  |-  ( (
ph 
<->  ps )  ->  ( ph  ->  ps ) )
1514imim2i 14 . . . . . . . 8  |-  ( ( x  =  A  -> 
( ph  <->  ps ) )  -> 
( x  =  A  ->  ( ph  ->  ps ) ) )
1615com23 75 . . . . . . 7  |-  ( ( x  =  A  -> 
( ph  <->  ps ) )  -> 
( ph  ->  ( x  =  A  ->  ps ) ) )
1716imp 420 . . . . . 6  |-  ( ( ( x  =  A  ->  ( ph  <->  ps )
)  /\  ph )  -> 
( x  =  A  ->  ps ) )
1817alanimi 1572 . . . . 5  |-  ( ( A. x ( x  =  A  ->  ( ph 
<->  ps ) )  /\  A. x ph )  ->  A. x ( x  =  A  ->  ps )
)
19183ad2ant2 980 . . . 4  |-  ( ( ( F/_ x A  /\  F/ x ps )  /\  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  /\  A. x ph )  /\  A  e.  _V )  ->  A. x
( x  =  A  ->  ps ) )
20 simp1r 983 . . . . 5  |-  ( ( ( F/_ x A  /\  F/ x ps )  /\  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  /\  A. x ph )  /\  A  e.  _V )  ->  F/ x ps )
21 19.23t 1820 . . . . 5  |-  ( F/ x ps  ->  ( A. x ( x  =  A  ->  ps )  <->  ( E. x  x  =  A  ->  ps )
) )
2220, 21syl 16 . . . 4  |-  ( ( ( F/_ x A  /\  F/ x ps )  /\  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  /\  A. x ph )  /\  A  e.  _V )  ->  ( A. x ( x  =  A  ->  ps )  <->  ( E. x  x  =  A  ->  ps )
) )
2319, 22mpbid 203 . . 3  |-  ( ( ( F/_ x A  /\  F/ x ps )  /\  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  /\  A. x ph )  /\  A  e.  _V )  ->  ( E. x  x  =  A  ->  ps ) )
2413, 23mpd 15 . 2  |-  ( ( ( F/_ x A  /\  F/ x ps )  /\  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  /\  A. x ph )  /\  A  e.  _V )  ->  ps )
251, 24syl3an3 1220 1  |-  ( ( ( F/_ x A  /\  F/ x ps )  /\  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  /\  A. x ph )  /\  A  e.  V )  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937   A.wal 1550   E.wex 1551   F/wnf 1554    = wceq 1653    e. wcel 1727   F/_wnfc 2565   _Vcvv 2962
This theorem is referenced by:  vtocldf  3009  riotasv2dOLD  6624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423
This theorem depends on definitions:  df-bi 179  df-an 362  df-3an 939  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-v 2964
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