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Theorem vtoclgft 2799
Description: Closed theorem form of vtoclgf 2807. (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 2760 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 elisset 2763 . . . . 5  |-  ( A  e.  _V  ->  E. z 
z  =  A )
323ad2ant3 1021 . . . 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 2332 . . . . . . 7  |-  F/ x F/_ x A
5 nfcvd 2330 . . . . . . . 8  |-  ( F/_ x A  ->  F/_ x
z )
6 id 19 . . . . . . . 8  |-  ( F/_ x A  ->  F/_ x A )
75, 6nfeqd 2344 . . . . . . 7  |-  ( F/_ x A  ->  F/ x  z  =  A )
8 eqeq1 2194 . . . . . . . 8  |-  ( z  =  x  ->  (
z  =  A  <->  x  =  A ) )
98a1i 9 . . . . . . 7  |-  ( F/_ x A  ->  ( z  =  x  ->  (
z  =  A  <->  x  =  A ) ) )
104, 7, 9cbvexd 1937 . . . . . 6  |-  ( F/_ x A  ->  ( E. z  z  =  A  <->  E. x  x  =  A ) )
1110ad2antrr 488 . . . . 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 1018 . . . 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 147 . . 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 118 . . . . . . . . 9  |-  ( (
ph 
<->  ps )  ->  ( ph  ->  ps ) )
1514imim2i 12 . . . . . . . 8  |-  ( ( x  =  A  -> 
( ph  <->  ps ) )  -> 
( x  =  A  ->  ( ph  ->  ps ) ) )
1615com23 78 . . . . . . 7  |-  ( ( x  =  A  -> 
( ph  <->  ps ) )  -> 
( ph  ->  ( x  =  A  ->  ps ) ) )
1716imp 124 . . . . . 6  |-  ( ( ( x  =  A  ->  ( ph  <->  ps )
)  /\  ph )  -> 
( x  =  A  ->  ps ) )
1817alanimi 1469 . . . . 5  |-  ( ( A. x ( x  =  A  ->  ( ph 
<->  ps ) )  /\  A. x ph )  ->  A. x ( x  =  A  ->  ps )
)
19183ad2ant2 1020 . . . 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 1023 . . . . 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 1687 . . . . 5  |-  ( F/ x ps  ->  ( A. x ( x  =  A  ->  ps )  <->  ( E. x  x  =  A  ->  ps )
) )
2220, 21syl 14 . . . 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 147 . . 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 13 . 2  |-  ( ( ( F/_ x A  /\  F/ x ps )  /\  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  /\  A. x ph )  /\  A  e.  _V )  ->  ps )
251, 24syl3an3 1283 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    /\ wa 104    <-> wb 105    /\ w3a 979   A.wal 1361    = wceq 1363   F/wnf 1470   E.wex 1502    e. wcel 2158   F/_wnfc 2316   _Vcvv 2749
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-3an 981  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751
This theorem is referenced by:  vtocldf  2800
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