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Theorem vtoclgft 2836
Description: Closed theorem form of vtoclgf 2844. (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 2798 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 elisset 2800 . . . . 5  |-  ( A  e.  _V  ->  E. z 
z  =  A )
323ad2ant3 978 . . . 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 2424 . . . . . . 7  |-  F/ x F/_ x A
5 nfcvd 2422 . . . . . . . 8  |-  ( F/_ x A  ->  F/_ x
z )
6 id 19 . . . . . . . 8  |-  ( F/_ x A  ->  F/_ x A )
75, 6nfeqd 2435 . . . . . . 7  |-  ( F/_ x A  ->  F/ x  z  =  A )
8 eqeq1 2291 . . . . . . . 8  |-  ( z  =  x  ->  (
z  =  A  <->  x  =  A ) )
98a1i 10 . . . . . . 7  |-  ( F/_ x A  ->  ( z  =  x  ->  (
z  =  A  <->  x  =  A ) ) )
104, 7, 9cbvexd 1951 . . . . . 6  |-  ( F/_ x A  ->  ( E. z  z  =  A  <->  E. x  x  =  A ) )
1110ad2antrr 706 . . . . 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 975 . . . 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 201 . . 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 178 . . . . . . . . 9  |-  ( (
ph 
<->  ps )  ->  ( ph  ->  ps ) )
1514imim2i 13 . . . . . . . 8  |-  ( ( x  =  A  -> 
( ph  <->  ps ) )  -> 
( x  =  A  ->  ( ph  ->  ps ) ) )
1615com23 72 . . . . . . 7  |-  ( ( x  =  A  -> 
( ph  <->  ps ) )  -> 
( ph  ->  ( x  =  A  ->  ps ) ) )
1716imp 418 . . . . . 6  |-  ( ( ( x  =  A  ->  ( ph  <->  ps )
)  /\  ph )  -> 
( x  =  A  ->  ps ) )
1817alanimi 1551 . . . . 5  |-  ( ( A. x ( x  =  A  ->  ( ph 
<->  ps ) )  /\  A. x ph )  ->  A. x ( x  =  A  ->  ps )
)
19183ad2ant2 977 . . . 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 980 . . . . 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 1798 . . . . 5  |-  ( F/ x ps  ->  ( A. x ( x  =  A  ->  ps )  <->  ( E. x  x  =  A  ->  ps )
) )
2220, 21syl 15 . . . 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 201 . . 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 14 . 2  |-  ( ( ( F/_ x A  /\  F/ x ps )  /\  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  /\  A. x ph )  /\  A  e.  _V )  ->  ps )
251, 24syl3an3 1217 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 176    /\ wa 358    /\ w3a 934   A.wal 1529   E.wex 1530   F/wnf 1533    = wceq 1625    e. wcel 1686   F/_wnfc 2408   _Vcvv 2790
This theorem is referenced by:  vtocldf  2837  iota2df  5245  riotasv2dOLD  6352
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-ex 1531  df-nf 1534  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-v 2792
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