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Theorem sbcabel 3239
Description: Interchange class substitution and class abstraction. (Contributed by NM, 5-Nov-2005.)
Hypothesis
Ref Expression
sbcabel.1  |-  F/_ x B
Assertion
Ref Expression
sbcabel  |-  ( A  e.  V  ->  ( [. A  /  x ]. { y  |  ph }  e.  B  <->  { y  |  [. A  /  x ]. ph }  e.  B
) )
Distinct variable groups:    y, A    x, y
Allowed substitution hints:    ph( x, y)    A( x)    B( x, y)    V( x, y)

Proof of Theorem sbcabel
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 elex 2965 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 sbcexg 3212 . . . 4  |-  ( A  e.  _V  ->  ( [. A  /  x ]. E. w ( w  =  { y  | 
ph }  /\  w  e.  B )  <->  E. w [. A  /  x ]. ( w  =  {
y  |  ph }  /\  w  e.  B
) ) )
3 sbcang 3205 . . . . . 6  |-  ( A  e.  _V  ->  ( [. A  /  x ]. ( w  =  {
y  |  ph }  /\  w  e.  B
)  <->  ( [. A  /  x ]. w  =  { y  |  ph }  /\  [. A  /  x ]. w  e.  B
) ) )
4 sbcalg 3210 . . . . . . . . 9  |-  ( A  e.  _V  ->  ( [. A  /  x ]. A. y ( y  e.  w  <->  ph )  <->  A. y [. A  /  x ]. ( y  e.  w  <->  ph ) ) )
5 sbcbig 3208 . . . . . . . . . . 11  |-  ( A  e.  _V  ->  ( [. A  /  x ]. ( y  e.  w  <->  ph )  <->  ( [. A  /  x ]. y  e.  w  <->  [. A  /  x ]. ph ) ) )
6 sbcg 3227 . . . . . . . . . . . 12  |-  ( A  e.  _V  ->  ( [. A  /  x ]. y  e.  w  <->  y  e.  w ) )
76bibi1d 312 . . . . . . . . . . 11  |-  ( A  e.  _V  ->  (
( [. A  /  x ]. y  e.  w  <->  [. A  /  x ]. ph )  <->  ( y  e.  w  <->  [. A  /  x ]. ph ) ) )
85, 7bitrd 246 . . . . . . . . . 10  |-  ( A  e.  _V  ->  ( [. A  /  x ]. ( y  e.  w  <->  ph )  <->  ( y  e.  w  <->  [. A  /  x ]. ph ) ) )
98albidv 1636 . . . . . . . . 9  |-  ( A  e.  _V  ->  ( A. y [. A  /  x ]. ( y  e.  w  <->  ph )  <->  A. y
( y  e.  w  <->  [. A  /  x ]. ph ) ) )
104, 9bitrd 246 . . . . . . . 8  |-  ( A  e.  _V  ->  ( [. A  /  x ]. A. y ( y  e.  w  <->  ph )  <->  A. y
( y  e.  w  <->  [. A  /  x ]. ph ) ) )
11 abeq2 2542 . . . . . . . . 9  |-  ( w  =  { y  | 
ph }  <->  A. y
( y  e.  w  <->  ph ) )
1211sbcbii 3217 . . . . . . . 8  |-  ( [. A  /  x ]. w  =  { y  |  ph } 
<-> 
[. A  /  x ]. A. y ( y  e.  w  <->  ph ) )
13 abeq2 2542 . . . . . . . 8  |-  ( w  =  { y  | 
[. A  /  x ]. ph }  <->  A. y
( y  e.  w  <->  [. A  /  x ]. ph ) )
1410, 12, 133bitr4g 281 . . . . . . 7  |-  ( A  e.  _V  ->  ( [. A  /  x ]. w  =  {
y  |  ph }  <->  w  =  { y  | 
[. A  /  x ]. ph } ) )
15 sbcabel.1 . . . . . . . . 9  |-  F/_ x B
1615nfcri 2567 . . . . . . . 8  |-  F/ x  w  e.  B
1716sbcgf 3225 . . . . . . 7  |-  ( A  e.  _V  ->  ( [. A  /  x ]. w  e.  B  <->  w  e.  B ) )
1814, 17anbi12d 693 . . . . . 6  |-  ( A  e.  _V  ->  (
( [. A  /  x ]. w  =  {
y  |  ph }  /\  [. A  /  x ]. w  e.  B
)  <->  ( w  =  { y  |  [. A  /  x ]. ph }  /\  w  e.  B
) ) )
193, 18bitrd 246 . . . . 5  |-  ( A  e.  _V  ->  ( [. A  /  x ]. ( w  =  {
y  |  ph }  /\  w  e.  B
)  <->  ( w  =  { y  |  [. A  /  x ]. ph }  /\  w  e.  B
) ) )
2019exbidv 1637 . . . 4  |-  ( A  e.  _V  ->  ( E. w [. A  /  x ]. ( w  =  { y  |  ph }  /\  w  e.  B
)  <->  E. w ( w  =  { y  | 
[. A  /  x ]. ph }  /\  w  e.  B ) ) )
212, 20bitrd 246 . . 3  |-  ( A  e.  _V  ->  ( [. A  /  x ]. E. w ( w  =  { y  | 
ph }  /\  w  e.  B )  <->  E. w
( w  =  {
y  |  [. A  /  x ]. ph }  /\  w  e.  B
) ) )
22 df-clel 2433 . . . 4  |-  ( { y  |  ph }  e.  B  <->  E. w ( w  =  { y  | 
ph }  /\  w  e.  B ) )
2322sbcbii 3217 . . 3  |-  ( [. A  /  x ]. {
y  |  ph }  e.  B  <->  [. A  /  x ]. E. w ( w  =  { y  | 
ph }  /\  w  e.  B ) )
24 df-clel 2433 . . 3  |-  ( { y  |  [. A  /  x ]. ph }  e.  B  <->  E. w ( w  =  { y  | 
[. A  /  x ]. ph }  /\  w  e.  B ) )
2521, 23, 243bitr4g 281 . 2  |-  ( A  e.  _V  ->  ( [. A  /  x ]. { y  |  ph }  e.  B  <->  { y  |  [. A  /  x ]. ph }  e.  B
) )
261, 25syl 16 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. { y  |  ph }  e.  B  <->  { y  |  [. A  /  x ]. ph }  e.  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360   A.wal 1550   E.wex 1551    = wceq 1653    e. wcel 1726   {cab 2423   F/_wnfc 2560   _Vcvv 2957   [.wsbc 3162
This theorem is referenced by:  csbexg  3262
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-v 2959  df-sbc 3163
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