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Theorem opelopabsb 4257
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 4255 . . . 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 109 . . . . . . . 8  |-  ( (
<. A ,  B >.  = 
<. u ,  v >.  /\  [ u  /  x ] [ v  /  y ] ph )  ->  <. A ,  B >.  =  <. u ,  v >. )
32eqcomd 2183 . . . . . . 7  |-  ( (
<. A ,  B >.  = 
<. u ,  v >.  /\  [ u  /  x ] [ v  /  y ] ph )  ->  <. u ,  v >.  =  <. A ,  B >. )
4 vex 2740 . . . . . . . 8  |-  u  e. 
_V
5 vex 2740 . . . . . . . 8  |-  v  e. 
_V
64, 5opth 4234 . . . . . . 7  |-  ( <.
u ,  v >.  =  <. A ,  B >.  <-> 
( u  =  A  /\  v  =  B ) )
73, 6sylib 122 . . . . . 6  |-  ( (
<. A ,  B >.  = 
<. u ,  v >.  /\  [ u  /  x ] [ v  /  y ] ph )  ->  (
u  =  A  /\  v  =  B )
)
872eximi 1601 . . . . 5  |-  ( E. u E. v (
<. A ,  B >.  = 
<. u ,  v >.  /\  [ u  /  x ] [ v  /  y ] ph )  ->  E. u E. v ( u  =  A  /\  v  =  B ) )
9 eeanv 1932 . . . . . 6  |-  ( E. u E. v ( u  =  A  /\  v  =  B )  <->  ( E. u  u  =  A  /\  E. v 
v  =  B ) )
10 isset 2743 . . . . . . 7  |-  ( A  e.  _V  <->  E. u  u  =  A )
11 isset 2743 . . . . . . 7  |-  ( B  e.  _V  <->  E. v 
v  =  B )
1210, 11anbi12i 460 . . . . . 6  |-  ( ( A  e.  _V  /\  B  e.  _V )  <->  ( E. u  u  =  A  /\  E. v 
v  =  B ) )
139, 12bitr4i 187 . . . . 5  |-  ( E. u E. v ( u  =  A  /\  v  =  B )  <->  ( A  e.  _V  /\  B  e.  _V )
)
148, 13sylib 122 . . . 4  |-  ( E. u E. v (
<. A ,  B >.  = 
<. u ,  v >.  /\  [ u  /  x ] [ v  /  y ] ph )  ->  ( A  e.  _V  /\  B  e.  _V ) )
151, 14sylbi 121 . . 3  |-  ( <. A ,  B >.  e. 
{ <. u ,  v
>.  |  [ u  /  x ] [ v  /  y ] ph }  ->  ( A  e. 
_V  /\  B  e.  _V ) )
16 nfv 1528 . . . 4  |-  F/ u ph
17 nfv 1528 . . . 4  |-  F/ v
ph
18 nfs1v 1939 . . . 4  |-  F/ x [ u  /  x ] [ v  /  y ] ph
19 nfs1v 1939 . . . . 5  |-  F/ y [ v  /  y ] ph
2019nfsbxy 1942 . . . 4  |-  F/ y [ u  /  x ] [ v  /  y ] ph
21 sbequ12 1771 . . . . 5  |-  ( y  =  v  ->  ( ph 
<->  [ v  /  y ] ph ) )
22 sbequ12 1771 . . . . 5  |-  ( x  =  u  ->  ( [ v  /  y ] ph  <->  [ u  /  x ] [ v  /  y ] ph ) )
2321, 22sylan9bbr 463 . . . 4  |-  ( ( x  =  u  /\  y  =  v )  ->  ( ph  <->  [ u  /  x ] [ v  /  y ] ph ) )
2416, 17, 18, 20, 23cbvopab 4071 . . 3  |-  { <. x ,  y >.  |  ph }  =  { <. u ,  v >.  |  [
u  /  x ] [ v  /  y ] ph }
2515, 24eleq2s 2272 . 2  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  ->  ( A  e.  _V  /\  B  e.  _V )
)
26 sbcex 2971 . . 3  |-  ( [. A  /  x ]. [. B  /  y ]. ph  ->  A  e.  _V )
27 spesbc 3048 . . . 4  |-  ( [. A  /  x ]. [. B  /  y ]. ph  ->  E. x [. B  / 
y ]. ph )
28 sbcex 2971 . . . . 5  |-  ( [. B  /  y ]. ph  ->  B  e.  _V )
2928exlimiv 1598 . . . 4  |-  ( E. x [. B  / 
y ]. ph  ->  B  e.  _V )
3027, 29syl 14 . . 3  |-  ( [. A  /  x ]. [. B  /  y ]. ph  ->  B  e.  _V )
3126, 30jca 306 . 2  |-  ( [. A  /  x ]. [. B  /  y ]. ph  ->  ( A  e.  _V  /\  B  e.  _V )
)
32 opeq1 3776 . . . . 5  |-  ( z  =  A  ->  <. z ,  w >.  =  <. A ,  w >. )
3332eleq1d 2246 . . . 4  |-  ( z  =  A  ->  ( <. z ,  w >.  e. 
{ <. x ,  y
>.  |  ph }  <->  <. A ,  w >.  e.  { <. x ,  y >.  |  ph } ) )
34 dfsbcq2 2965 . . . 4  |-  ( z  =  A  ->  ( [ z  /  x ] [ w  /  y ] ph  <->  [. A  /  x ]. [ w  /  y ] ph ) )
3533, 34bibi12d 235 . . 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 3777 . . . . 5  |-  ( w  =  B  ->  <. A ,  w >.  =  <. A ,  B >. )
3736eleq1d 2246 . . . 4  |-  ( w  =  B  ->  ( <. A ,  w >.  e. 
{ <. x ,  y
>.  |  ph }  <->  <. A ,  B >.  e.  { <. x ,  y >.  |  ph } ) )
38 dfsbcq2 2965 . . . . 5  |-  ( w  =  B  ->  ( [ w  /  y ] ph  <->  [. B  /  y ]. ph ) )
3938sbcbidv 3021 . . . 4  |-  ( w  =  B  ->  ( [. A  /  x ]. [ w  /  y ] ph  <->  [. A  /  x ]. [. B  /  y ]. ph ) )
4037, 39bibi12d 235 . . 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 4069 . . . . . 6  |-  F/_ x { <. x ,  y
>.  |  ph }
4241nfel2 2332 . . . . 5  |-  F/ x <. z ,  w >.  e. 
{ <. x ,  y
>.  |  ph }
43 nfs1v 1939 . . . . 5  |-  F/ x [ z  /  x ] [ w  /  y ] ph
4442, 43nfbi 1589 . . . 4  |-  F/ x
( <. z ,  w >.  e.  { <. x ,  y >.  |  ph } 
<->  [ z  /  x ] [ w  /  y ] ph )
45 opeq1 3776 . . . . . 6  |-  ( x  =  z  ->  <. x ,  w >.  =  <. z ,  w >. )
4645eleq1d 2246 . . . . 5  |-  ( x  =  z  ->  ( <. x ,  w >.  e. 
{ <. x ,  y
>.  |  ph }  <->  <. z ,  w >.  e.  { <. x ,  y >.  |  ph } ) )
47 sbequ12 1771 . . . . 5  |-  ( x  =  z  ->  ( [ w  /  y ] ph  <->  [ z  /  x ] [ w  /  y ] ph ) )
4846, 47bibi12d 235 . . . 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 4070 . . . . . . 7  |-  F/_ y { <. x ,  y
>.  |  ph }
5049nfel2 2332 . . . . . 6  |-  F/ y
<. x ,  w >.  e. 
{ <. x ,  y
>.  |  ph }
51 nfs1v 1939 . . . . . 6  |-  F/ y [ w  /  y ] ph
5250, 51nfbi 1589 . . . . 5  |-  F/ y ( <. x ,  w >.  e.  { <. x ,  y >.  |  ph } 
<->  [ w  /  y ] ph )
53 opeq2 3777 . . . . . . 7  |-  ( y  =  w  ->  <. x ,  y >.  =  <. x ,  w >. )
5453eleq1d 2246 . . . . . 6  |-  ( y  =  w  ->  ( <. x ,  y >.  e.  { <. x ,  y
>.  |  ph }  <->  <. x ,  w >.  e.  { <. x ,  y >.  |  ph } ) )
55 sbequ12 1771 . . . . . 6  |-  ( y  =  w  ->  ( ph 
<->  [ w  /  y ] ph ) )
5654, 55bibi12d 235 . . . . 5  |-  ( y  =  w  ->  (
( <. x ,  y
>.  e.  { <. x ,  y >.  |  ph } 
<-> 
ph )  <->  ( <. x ,  w >.  e.  { <. x ,  y >.  |  ph }  <->  [ w  /  y ] ph ) ) )
57 opabid 4254 . . . . 5  |-  ( <.
x ,  y >.  e.  { <. x ,  y
>.  |  ph }  <->  ph )
5852, 56, 57chvar 1757 . . . 4  |-  ( <.
x ,  w >.  e. 
{ <. x ,  y
>.  |  ph }  <->  [ w  /  y ] ph )
5944, 48, 58chvar 1757 . . 3  |-  ( <.
z ,  w >.  e. 
{ <. x ,  y
>.  |  ph }  <->  [ z  /  x ] [ w  /  y ] ph )
6035, 40, 59vtocl2g 2801 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ph } 
<-> 
[. A  /  x ]. [. B  /  y ]. ph ) )
6125, 31, 60pm5.21nii 704 1  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  <->  [. A  /  x ]. [. B  / 
y ]. ph )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105    = wceq 1353   E.wex 1492   [wsb 1762    e. wcel 2148   _Vcvv 2737   [.wsbc 2962   <.cop 3594   {copab 4060
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-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-opab 4062
This theorem is referenced by:  brabsb  4258  opelopabaf  4270  opelopabf  4271  difopab  4756  isarep1  5298
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