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Theorem opelopabsb 3994
Description: The law of concretion in terms of substitutions. (Contributed by NM, 30-Sep-2002.) (Revised by Mario Carneiro, 18-Nov-2016.)
Assertion
Ref Expression
opelopabsb  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  <->  [. A  /  x ]. [. B  / 
y ]. ph )
Distinct variable groups:    x, y    x, B
Allowed substitution hints:    ph( x, y)    A( x, y)    B( y)

Proof of Theorem opelopabsb
Dummy variables  v  u  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elopab 3992 . . . 4  |-  ( <. A ,  B >.  e. 
{ <. u ,  v
>.  |  [ u  /  x ] [ v  /  y ] ph } 
<->  E. u E. v
( <. A ,  B >.  =  <. u ,  v
>.  /\  [ u  /  x ] [ v  / 
y ] ph )
)
2 simpl 102 . . . . . . . 8  |-  ( (
<. A ,  B >.  = 
<. u ,  v >.  /\  [ u  /  x ] [ v  /  y ] ph )  ->  <. A ,  B >.  =  <. u ,  v >. )
32eqcomd 2045 . . . . . . 7  |-  ( (
<. A ,  B >.  = 
<. u ,  v >.  /\  [ u  /  x ] [ v  /  y ] ph )  ->  <. u ,  v >.  =  <. A ,  B >. )
4 vex 2557 . . . . . . . 8  |-  u  e. 
_V
5 vex 2557 . . . . . . . 8  |-  v  e. 
_V
64, 5opth 3971 . . . . . . 7  |-  ( <.
u ,  v >.  =  <. A ,  B >.  <-> 
( u  =  A  /\  v  =  B ) )
73, 6sylib 127 . . . . . 6  |-  ( (
<. A ,  B >.  = 
<. u ,  v >.  /\  [ u  /  x ] [ v  /  y ] ph )  ->  (
u  =  A  /\  v  =  B )
)
872eximi 1492 . . . . 5  |-  ( E. u E. v (
<. A ,  B >.  = 
<. u ,  v >.  /\  [ u  /  x ] [ v  /  y ] ph )  ->  E. u E. v ( u  =  A  /\  v  =  B ) )
9 eeanv 1807 . . . . . 6  |-  ( E. u E. v ( u  =  A  /\  v  =  B )  <->  ( E. u  u  =  A  /\  E. v 
v  =  B ) )
10 isset 2558 . . . . . . 7  |-  ( A  e.  _V  <->  E. u  u  =  A )
11 isset 2558 . . . . . . 7  |-  ( B  e.  _V  <->  E. v 
v  =  B )
1210, 11anbi12i 433 . . . . . 6  |-  ( ( A  e.  _V  /\  B  e.  _V )  <->  ( E. u  u  =  A  /\  E. v 
v  =  B ) )
139, 12bitr4i 176 . . . . 5  |-  ( E. u E. v ( u  =  A  /\  v  =  B )  <->  ( A  e.  _V  /\  B  e.  _V )
)
148, 13sylib 127 . . . 4  |-  ( E. u E. v (
<. A ,  B >.  = 
<. u ,  v >.  /\  [ u  /  x ] [ v  /  y ] ph )  ->  ( A  e.  _V  /\  B  e.  _V ) )
151, 14sylbi 114 . . 3  |-  ( <. A ,  B >.  e. 
{ <. u ,  v
>.  |  [ u  /  x ] [ v  /  y ] ph }  ->  ( A  e. 
_V  /\  B  e.  _V ) )
16 nfv 1421 . . . 4  |-  F/ u ph
17 nfv 1421 . . . 4  |-  F/ v
ph
18 nfs1v 1815 . . . 4  |-  F/ x [ u  /  x ] [ v  /  y ] ph
19 nfs1v 1815 . . . . 5  |-  F/ y [ v  /  y ] ph
2019nfsbxy 1818 . . . 4  |-  F/ y [ u  /  x ] [ v  /  y ] ph
21 sbequ12 1654 . . . . 5  |-  ( y  =  v  ->  ( ph 
<->  [ v  /  y ] ph ) )
22 sbequ12 1654 . . . . 5  |-  ( x  =  u  ->  ( [ v  /  y ] ph  <->  [ u  /  x ] [ v  /  y ] ph ) )
2321, 22sylan9bbr 436 . . . 4  |-  ( ( x  =  u  /\  y  =  v )  ->  ( ph  <->  [ u  /  x ] [ v  /  y ] ph ) )
2416, 17, 18, 20, 23cbvopab 3825 . . 3  |-  { <. x ,  y >.  |  ph }  =  { <. u ,  v >.  |  [
u  /  x ] [ v  /  y ] ph }
2515, 24eleq2s 2132 . 2  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  ->  ( A  e.  _V  /\  B  e.  _V )
)
26 sbcex 2769 . . 3  |-  ( [. A  /  x ]. [. B  /  y ]. ph  ->  A  e.  _V )
27 spesbc 2840 . . . 4  |-  ( [. A  /  x ]. [. B  /  y ]. ph  ->  E. x [. B  / 
y ]. ph )
28 sbcex 2769 . . . . 5  |-  ( [. B  /  y ]. ph  ->  B  e.  _V )
2928exlimiv 1489 . . . 4  |-  ( E. x [. B  / 
y ]. ph  ->  B  e.  _V )
3027, 29syl 14 . . 3  |-  ( [. A  /  x ]. [. B  /  y ]. ph  ->  B  e.  _V )
3126, 30jca 290 . 2  |-  ( [. A  /  x ]. [. B  /  y ]. ph  ->  ( A  e.  _V  /\  B  e.  _V )
)
32 opeq1 3546 . . . . 5  |-  ( z  =  A  ->  <. z ,  w >.  =  <. A ,  w >. )
3332eleq1d 2106 . . . 4  |-  ( z  =  A  ->  ( <. z ,  w >.  e. 
{ <. x ,  y
>.  |  ph }  <->  <. A ,  w >.  e.  { <. x ,  y >.  |  ph } ) )
34 dfsbcq2 2764 . . . 4  |-  ( z  =  A  ->  ( [ z  /  x ] [ w  /  y ] ph  <->  [. A  /  x ]. [ w  /  y ] ph ) )
3533, 34bibi12d 224 . . 3  |-  ( z  =  A  ->  (
( <. z ,  w >.  e.  { <. x ,  y >.  |  ph } 
<->  [ z  /  x ] [ w  /  y ] ph )  <->  ( <. A ,  w >.  e.  { <. x ,  y >.  |  ph }  <->  [. A  /  x ]. [ w  / 
y ] ph )
) )
36 opeq2 3547 . . . . 5  |-  ( w  =  B  ->  <. A ,  w >.  =  <. A ,  B >. )
3736eleq1d 2106 . . . 4  |-  ( w  =  B  ->  ( <. A ,  w >.  e. 
{ <. x ,  y
>.  |  ph }  <->  <. A ,  B >.  e.  { <. x ,  y >.  |  ph } ) )
38 dfsbcq2 2764 . . . . 5  |-  ( w  =  B  ->  ( [ w  /  y ] ph  <->  [. B  /  y ]. ph ) )
3938sbcbidv 2814 . . . 4  |-  ( w  =  B  ->  ( [. A  /  x ]. [ w  /  y ] ph  <->  [. A  /  x ]. [. B  /  y ]. ph ) )
4037, 39bibi12d 224 . . 3  |-  ( w  =  B  ->  (
( <. A ,  w >.  e.  { <. x ,  y >.  |  ph } 
<-> 
[. A  /  x ]. [ w  /  y ] ph )  <->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ph }  <->  [. A  /  x ]. [. B  / 
y ]. ph ) ) )
41 nfopab1 3823 . . . . . 6  |-  F/_ x { <. x ,  y
>.  |  ph }
4241nfel2 2190 . . . . 5  |-  F/ x <. z ,  w >.  e. 
{ <. x ,  y
>.  |  ph }
43 nfs1v 1815 . . . . 5  |-  F/ x [ z  /  x ] [ w  /  y ] ph
4442, 43nfbi 1481 . . . 4  |-  F/ x
( <. z ,  w >.  e.  { <. x ,  y >.  |  ph } 
<->  [ z  /  x ] [ w  /  y ] ph )
45 opeq1 3546 . . . . . 6  |-  ( x  =  z  ->  <. x ,  w >.  =  <. z ,  w >. )
4645eleq1d 2106 . . . . 5  |-  ( x  =  z  ->  ( <. x ,  w >.  e. 
{ <. x ,  y
>.  |  ph }  <->  <. z ,  w >.  e.  { <. x ,  y >.  |  ph } ) )
47 sbequ12 1654 . . . . 5  |-  ( x  =  z  ->  ( [ w  /  y ] ph  <->  [ z  /  x ] [ w  /  y ] ph ) )
4846, 47bibi12d 224 . . . 4  |-  ( x  =  z  ->  (
( <. x ,  w >.  e.  { <. x ,  y >.  |  ph } 
<->  [ w  /  y ] ph )  <->  ( <. z ,  w >.  e.  { <. x ,  y >.  |  ph }  <->  [ z  /  x ] [ w  /  y ] ph ) ) )
49 nfopab2 3824 . . . . . . 7  |-  F/_ y { <. x ,  y
>.  |  ph }
5049nfel2 2190 . . . . . 6  |-  F/ y
<. x ,  w >.  e. 
{ <. x ,  y
>.  |  ph }
51 nfs1v 1815 . . . . . 6  |-  F/ y [ w  /  y ] ph
5250, 51nfbi 1481 . . . . 5  |-  F/ y ( <. x ,  w >.  e.  { <. x ,  y >.  |  ph } 
<->  [ w  /  y ] ph )
53 opeq2 3547 . . . . . . 7  |-  ( y  =  w  ->  <. x ,  y >.  =  <. x ,  w >. )
5453eleq1d 2106 . . . . . 6  |-  ( y  =  w  ->  ( <. x ,  y >.  e.  { <. x ,  y
>.  |  ph }  <->  <. x ,  w >.  e.  { <. x ,  y >.  |  ph } ) )
55 sbequ12 1654 . . . . . 6  |-  ( y  =  w  ->  ( ph 
<->  [ w  /  y ] ph ) )
5654, 55bibi12d 224 . . . . 5  |-  ( y  =  w  ->  (
( <. x ,  y
>.  e.  { <. x ,  y >.  |  ph } 
<-> 
ph )  <->  ( <. x ,  w >.  e.  { <. x ,  y >.  |  ph }  <->  [ w  /  y ] ph ) ) )
57 opabid 3991 . . . . 5  |-  ( <.
x ,  y >.  e.  { <. x ,  y
>.  |  ph }  <->  ph )
5852, 56, 57chvar 1640 . . . 4  |-  ( <.
x ,  w >.  e. 
{ <. x ,  y
>.  |  ph }  <->  [ w  /  y ] ph )
5944, 48, 58chvar 1640 . . 3  |-  ( <.
z ,  w >.  e. 
{ <. x ,  y
>.  |  ph }  <->  [ z  /  x ] [ w  /  y ] ph )
6035, 40, 59vtocl2g 2614 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ph } 
<-> 
[. A  /  x ]. [. B  /  y ]. ph ) )
6125, 31, 60pm5.21nii 620 1  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  <->  [. A  /  x ]. [. B  / 
y ]. ph )
Colors of variables: wff set class
Syntax hints:    /\ wa 97    <-> wb 98    = wceq 1243   E.wex 1381    e. wcel 1393   [wsb 1645   _Vcvv 2554   [.wsbc 2761   <.cop 3375   {copab 3814
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3872  ax-pow 3924  ax-pr 3941
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2309  df-v 2556  df-sbc 2762  df-un 2919  df-in 2921  df-ss 2928  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-opab 3816
This theorem is referenced by:  brabsb  3995  opelopabaf  4007  opelopabf  4008  difopab  4447  isarep1  4963
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