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Theorem vtoclgft 3129
Description: Closed theorem form of vtoclgf 3137. (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 3089 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 elisset 3091 . . . . 5  |-  ( A  e.  _V  ->  E. z 
z  =  A )
323ad2ant3 1028 . . . 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 2583 . . . . . . 7  |-  F/ x F/_ x A
5 nfcvd 2581 . . . . . . . 8  |-  ( F/_ x A  ->  F/_ x
z )
6 id 22 . . . . . . . 8  |-  ( F/_ x A  ->  F/_ x A )
75, 6nfeqd 2587 . . . . . . 7  |-  ( F/_ x A  ->  F/ x  z  =  A )
8 eqeq1 2426 . . . . . . . 8  |-  ( z  =  x  ->  (
z  =  A  <->  x  =  A ) )
98a1i 11 . . . . . . 7  |-  ( F/_ x A  ->  ( z  =  x  ->  (
z  =  A  <->  x  =  A ) ) )
104, 7, 9cbvexd 2084 . . . . . 6  |-  ( F/_ x A  ->  ( E. z  z  =  A  <->  E. x  x  =  A ) )
1110ad2antrr 730 . . . . 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 1025 . . . 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 213 . . 3  |-  ( ( ( F/_ x A  /\  F/ x ps )  /\  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  /\  A. x ph )  /\  A  e.  _V )  ->  E. x  x  =  A )
14 biimp 196 . . . . . . . . 9  |-  ( (
ph 
<->  ps )  ->  ( ph  ->  ps ) )
1514imim2i 16 . . . . . . . 8  |-  ( ( x  =  A  -> 
( ph  <->  ps ) )  -> 
( x  =  A  ->  ( ph  ->  ps ) ) )
1615com23 81 . . . . . . 7  |-  ( ( x  =  A  -> 
( ph  <->  ps ) )  -> 
( ph  ->  ( x  =  A  ->  ps ) ) )
1716imp 430 . . . . . 6  |-  ( ( ( x  =  A  ->  ( ph  <->  ps )
)  /\  ph )  -> 
( x  =  A  ->  ps ) )
1817alanimi 1682 . . . . 5  |-  ( ( A. x ( x  =  A  ->  ( ph 
<->  ps ) )  /\  A. x ph )  ->  A. x ( x  =  A  ->  ps )
)
19183ad2ant2 1027 . . . 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 1030 . . . . 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 1968 . . . . 5  |-  ( F/ x ps  ->  ( A. x ( x  =  A  ->  ps )  <->  ( E. x  x  =  A  ->  ps )
) )
2220, 21syl 17 . . . 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 213 . . 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 1299 1  |-  ( ( ( F/_ x A  /\  F/ x ps )  /\  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  /\  A. x ph )  /\  A  e.  V )  ->  ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982   A.wal 1435    = wceq 1437   E.wex 1657   F/wnf 1661    e. wcel 1872   F/_wnfc 2566   _Vcvv 3080
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-v 3082
This theorem is referenced by:  vtocldf  3130  bj-vtoclgfALT  31593
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