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Theorem intab 3858
Description: The intersection of a special case of a class abstraction. 
y may be free in  ph and  A, which can be thought of a  ph ( y ) and  A ( y ). (Contributed by NM, 28-Jul-2006.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)
Hypotheses
Ref Expression
intab.1  |-  A  e. 
_V
intab.2  |-  { x  |  E. y ( ph  /\  x  =  A ) }  e.  _V
Assertion
Ref Expression
intab  |-  |^| { x  |  A. y ( ph  ->  A  e.  x ) }  =  { x  |  E. y ( ph  /\  x  =  A ) }
Distinct variable groups:    x, A    ph, x    x, y
Allowed substitution hints:    ph( y)    A( y)

Proof of Theorem intab
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2177 . . . . . . . . . 10  |-  ( z  =  x  ->  (
z  =  A  <->  x  =  A ) )
21anbi2d 461 . . . . . . . . 9  |-  ( z  =  x  ->  (
( ph  /\  z  =  A )  <->  ( ph  /\  x  =  A ) ) )
32exbidv 1818 . . . . . . . 8  |-  ( z  =  x  ->  ( E. y ( ph  /\  z  =  A )  <->  E. y ( ph  /\  x  =  A )
) )
43cbvabv 2295 . . . . . . 7  |-  { z  |  E. y (
ph  /\  z  =  A ) }  =  { x  |  E. y ( ph  /\  x  =  A ) }
5 intab.2 . . . . . . 7  |-  { x  |  E. y ( ph  /\  x  =  A ) }  e.  _V
64, 5eqeltri 2243 . . . . . 6  |-  { z  |  E. y (
ph  /\  z  =  A ) }  e.  _V
7 nfe1 1489 . . . . . . . . 9  |-  F/ y E. y ( ph  /\  z  =  A )
87nfab 2317 . . . . . . . 8  |-  F/_ y { z  |  E. y ( ph  /\  z  =  A ) }
98nfeq2 2324 . . . . . . 7  |-  F/ y  x  =  { z  |  E. y (
ph  /\  z  =  A ) }
10 eleq2 2234 . . . . . . . 8  |-  ( x  =  { z  |  E. y ( ph  /\  z  =  A ) }  ->  ( A  e.  x  <->  A  e.  { z  |  E. y (
ph  /\  z  =  A ) } ) )
1110imbi2d 229 . . . . . . 7  |-  ( x  =  { z  |  E. y ( ph  /\  z  =  A ) }  ->  ( ( ph  ->  A  e.  x
)  <->  ( ph  ->  A  e.  { z  |  E. y ( ph  /\  z  =  A ) } ) ) )
129, 11albid 1608 . . . . . 6  |-  ( x  =  { z  |  E. y ( ph  /\  z  =  A ) }  ->  ( A. y ( ph  ->  A  e.  x )  <->  A. y
( ph  ->  A  e. 
{ z  |  E. y ( ph  /\  z  =  A ) } ) ) )
136, 12elab 2874 . . . . 5  |-  ( { z  |  E. y
( ph  /\  z  =  A ) }  e.  { x  |  A. y
( ph  ->  A  e.  x ) }  <->  A. y
( ph  ->  A  e. 
{ z  |  E. y ( ph  /\  z  =  A ) } ) )
14 19.8a 1583 . . . . . . . . 9  |-  ( (
ph  /\  z  =  A )  ->  E. y
( ph  /\  z  =  A ) )
1514ex 114 . . . . . . . 8  |-  ( ph  ->  ( z  =  A  ->  E. y ( ph  /\  z  =  A ) ) )
1615alrimiv 1867 . . . . . . 7  |-  ( ph  ->  A. z ( z  =  A  ->  E. y
( ph  /\  z  =  A ) ) )
17 intab.1 . . . . . . . 8  |-  A  e. 
_V
1817sbc6 2980 . . . . . . 7  |-  ( [. A  /  z ]. E. y ( ph  /\  z  =  A )  <->  A. z ( z  =  A  ->  E. y
( ph  /\  z  =  A ) ) )
1916, 18sylibr 133 . . . . . 6  |-  ( ph  ->  [. A  /  z ]. E. y ( ph  /\  z  =  A ) )
20 df-sbc 2956 . . . . . 6  |-  ( [. A  /  z ]. E. y ( ph  /\  z  =  A )  <->  A  e.  { z  |  E. y ( ph  /\  z  =  A ) } )
2119, 20sylib 121 . . . . 5  |-  ( ph  ->  A  e.  { z  |  E. y (
ph  /\  z  =  A ) } )
2213, 21mpgbir 1446 . . . 4  |-  { z  |  E. y (
ph  /\  z  =  A ) }  e.  { x  |  A. y
( ph  ->  A  e.  x ) }
23 intss1 3844 . . . 4  |-  ( { z  |  E. y
( ph  /\  z  =  A ) }  e.  { x  |  A. y
( ph  ->  A  e.  x ) }  ->  |^|
{ x  |  A. y ( ph  ->  A  e.  x ) } 
C_  { z  |  E. y ( ph  /\  z  =  A ) } )
2422, 23ax-mp 5 . . 3  |-  |^| { x  |  A. y ( ph  ->  A  e.  x ) }  C_  { z  |  E. y ( ph  /\  z  =  A ) }
25 19.29r 1614 . . . . . . . 8  |-  ( ( E. y ( ph  /\  z  =  A )  /\  A. y (
ph  ->  A  e.  x
) )  ->  E. y
( ( ph  /\  z  =  A )  /\  ( ph  ->  A  e.  x ) ) )
26 simplr 525 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  =  A )  /\  ( ph  ->  A  e.  x
) )  ->  z  =  A )
27 pm3.35 345 . . . . . . . . . . 11  |-  ( (
ph  /\  ( ph  ->  A  e.  x ) )  ->  A  e.  x )
2827adantlr 474 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  =  A )  /\  ( ph  ->  A  e.  x
) )  ->  A  e.  x )
2926, 28eqeltrd 2247 . . . . . . . . 9  |-  ( ( ( ph  /\  z  =  A )  /\  ( ph  ->  A  e.  x
) )  ->  z  e.  x )
3029exlimiv 1591 . . . . . . . 8  |-  ( E. y ( ( ph  /\  z  =  A )  /\  ( ph  ->  A  e.  x ) )  ->  z  e.  x
)
3125, 30syl 14 . . . . . . 7  |-  ( ( E. y ( ph  /\  z  =  A )  /\  A. y (
ph  ->  A  e.  x
) )  ->  z  e.  x )
3231ex 114 . . . . . 6  |-  ( E. y ( ph  /\  z  =  A )  ->  ( A. y (
ph  ->  A  e.  x
)  ->  z  e.  x ) )
3332alrimiv 1867 . . . . 5  |-  ( E. y ( ph  /\  z  =  A )  ->  A. x ( A. y ( ph  ->  A  e.  x )  -> 
z  e.  x ) )
34 vex 2733 . . . . . 6  |-  z  e. 
_V
3534elintab 3840 . . . . 5  |-  ( z  e.  |^| { x  | 
A. y ( ph  ->  A  e.  x ) }  <->  A. x ( A. y ( ph  ->  A  e.  x )  -> 
z  e.  x ) )
3633, 35sylibr 133 . . . 4  |-  ( E. y ( ph  /\  z  =  A )  ->  z  e.  |^| { x  |  A. y ( ph  ->  A  e.  x ) } )
3736abssi 3222 . . 3  |-  { z  |  E. y (
ph  /\  z  =  A ) }  C_  |^|
{ x  |  A. y ( ph  ->  A  e.  x ) }
3824, 37eqssi 3163 . 2  |-  |^| { x  |  A. y ( ph  ->  A  e.  x ) }  =  { z  |  E. y (
ph  /\  z  =  A ) }
3938, 4eqtri 2191 1  |-  |^| { x  |  A. y ( ph  ->  A  e.  x ) }  =  { x  |  E. y ( ph  /\  x  =  A ) }
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   A.wal 1346    = wceq 1348   E.wex 1485    e. wcel 2141   {cab 2156   _Vcvv 2730   [.wsbc 2955    C_ wss 3121   |^|cint 3829
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-sbc 2956  df-in 3127  df-ss 3134  df-int 3830
This theorem is referenced by: (None)
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