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Theorem sbcabel 3102
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 2830 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 sbcexg 3075 . . . 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 3068 . . . . . 6  |-  ( A  e.  _V  ->  ( [. A  /  x ]. ( w  =  {
y  |  ph }  /\  w  e.  B
)  <->  ( [. A  /  x ]. w  =  { y  |  ph }  /\  [. A  /  x ]. w  e.  B
) ) )
4 abeq2 2421 . . . . . . . . . 10  |-  ( w  =  { y  | 
ph }  <->  A. y
( y  e.  w  <->  ph ) )
54sbcbii 3080 . . . . . . . . 9  |-  ( [. A  /  x ]. w  =  { y  |  ph } 
<-> 
[. A  /  x ]. A. y ( y  e.  w  <->  ph ) )
6 sbcalg 3073 . . . . . . . . . 10  |-  ( A  e.  _V  ->  ( [. A  /  x ]. A. y ( y  e.  w  <->  ph )  <->  A. y [. A  /  x ]. ( y  e.  w  <->  ph ) ) )
7 sbcbig 3071 . . . . . . . . . . . 12  |-  ( A  e.  _V  ->  ( [. A  /  x ]. ( y  e.  w  <->  ph )  <->  ( [. A  /  x ]. y  e.  w  <->  [. A  /  x ]. ph ) ) )
8 sbcg 3090 . . . . . . . . . . . . 13  |-  ( A  e.  _V  ->  ( [. A  /  x ]. y  e.  w  <->  y  e.  w ) )
98bibi1d 310 . . . . . . . . . . . 12  |-  ( A  e.  _V  ->  (
( [. A  /  x ]. y  e.  w  <->  [. A  /  x ]. ph )  <->  ( y  e.  w  <->  [. A  /  x ]. ph ) ) )
107, 9bitrd 244 . . . . . . . . . . 11  |-  ( A  e.  _V  ->  ( [. A  /  x ]. ( y  e.  w  <->  ph )  <->  ( y  e.  w  <->  [. A  /  x ]. ph ) ) )
1110albidv 1616 . . . . . . . . . 10  |-  ( A  e.  _V  ->  ( A. y [. A  /  x ]. ( y  e.  w  <->  ph )  <->  A. y
( y  e.  w  <->  [. A  /  x ]. ph ) ) )
126, 11bitrd 244 . . . . . . . . 9  |-  ( A  e.  _V  ->  ( [. A  /  x ]. A. y ( y  e.  w  <->  ph )  <->  A. y
( y  e.  w  <->  [. A  /  x ]. ph ) ) )
135, 12syl5bb 248 . . . . . . . 8  |-  ( A  e.  _V  ->  ( [. A  /  x ]. w  =  {
y  |  ph }  <->  A. y ( y  e.  w  <->  [. A  /  x ]. ph ) ) )
14 abeq2 2421 . . . . . . . 8  |-  ( w  =  { y  | 
[. A  /  x ]. ph }  <->  A. y
( y  e.  w  <->  [. A  /  x ]. ph ) )
1513, 14syl6bbr 254 . . . . . . 7  |-  ( A  e.  _V  ->  ( [. A  /  x ]. w  =  {
y  |  ph }  <->  w  =  { y  | 
[. A  /  x ]. ph } ) )
16 sbcabel.1 . . . . . . . . 9  |-  F/_ x B
1716nfcri 2446 . . . . . . . 8  |-  F/ x  w  e.  B
1817sbcgf 3088 . . . . . . 7  |-  ( A  e.  _V  ->  ( [. A  /  x ]. w  e.  B  <->  w  e.  B ) )
1915, 18anbi12d 691 . . . . . 6  |-  ( A  e.  _V  ->  (
( [. A  /  x ]. w  =  {
y  |  ph }  /\  [. A  /  x ]. w  e.  B
)  <->  ( w  =  { y  |  [. A  /  x ]. ph }  /\  w  e.  B
) ) )
203, 19bitrd 244 . . . . 5  |-  ( A  e.  _V  ->  ( [. A  /  x ]. ( w  =  {
y  |  ph }  /\  w  e.  B
)  <->  ( w  =  { y  |  [. A  /  x ]. ph }  /\  w  e.  B
) ) )
2120exbidv 1617 . . . 4  |-  ( A  e.  _V  ->  ( E. w [. A  /  x ]. ( w  =  { y  |  ph }  /\  w  e.  B
)  <->  E. w ( w  =  { y  | 
[. A  /  x ]. ph }  /\  w  e.  B ) ) )
222, 21bitrd 244 . . 3  |-  ( A  e.  _V  ->  ( [. A  /  x ]. E. w ( w  =  { y  | 
ph }  /\  w  e.  B )  <->  E. w
( w  =  {
y  |  [. A  /  x ]. ph }  /\  w  e.  B
) ) )
23 df-clel 2312 . . . 4  |-  ( { y  |  ph }  e.  B  <->  E. w ( w  =  { y  | 
ph }  /\  w  e.  B ) )
2423sbcbii 3080 . . 3  |-  ( [. A  /  x ]. {
y  |  ph }  e.  B  <->  [. A  /  x ]. E. w ( w  =  { y  | 
ph }  /\  w  e.  B ) )
25 df-clel 2312 . . 3  |-  ( { y  |  [. A  /  x ]. ph }  e.  B  <->  E. w ( w  =  { y  | 
[. A  /  x ]. ph }  /\  w  e.  B ) )
2622, 24, 253bitr4g 279 . 2  |-  ( A  e.  _V  ->  ( [. A  /  x ]. { y  |  ph }  e.  B  <->  { y  |  [. A  /  x ]. ph }  e.  B
) )
271, 26syl 15 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 176    /\ wa 358   A.wal 1531   E.wex 1532    = wceq 1633    e. wcel 1701   {cab 2302   F/_wnfc 2439   _Vcvv 2822   [.wsbc 3025
This theorem is referenced by:  csbexg  3125
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-v 2824  df-sbc 3026
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