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Theorem intab 3892
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 ). Typically, abrexex2 5780 or abexssex 5781 can be used to satisfy the second hypothesis. (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 2289 . . . . . . . . . 10  |-  ( z  =  x  ->  (
z  =  A  <->  x  =  A ) )
21anbi2d 684 . . . . . . . . 9  |-  ( z  =  x  ->  (
( ph  /\  z  =  A )  <->  ( ph  /\  x  =  A ) ) )
32exbidv 1612 . . . . . . . 8  |-  ( z  =  x  ->  ( E. y ( ph  /\  z  =  A )  <->  E. y ( ph  /\  x  =  A )
) )
43cbvabv 2402 . . . . . . 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 2353 . . . . . 6  |-  { z  |  E. y (
ph  /\  z  =  A ) }  e.  _V
7 nfe1 1706 . . . . . . . . 9  |-  F/ y E. y ( ph  /\  z  =  A )
87nfab 2423 . . . . . . . 8  |-  F/_ y { z  |  E. y ( ph  /\  z  =  A ) }
98nfeq2 2430 . . . . . . 7  |-  F/ y  x  =  { z  |  E. y (
ph  /\  z  =  A ) }
10 eleq2 2344 . . . . . . . 8  |-  ( x  =  { z  |  E. y ( ph  /\  z  =  A ) }  ->  ( A  e.  x  <->  A  e.  { z  |  E. y (
ph  /\  z  =  A ) } ) )
1110imbi2d 307 . . . . . . 7  |-  ( x  =  { z  |  E. y ( ph  /\  z  =  A ) }  ->  ( ( ph  ->  A  e.  x
)  <->  ( ph  ->  A  e.  { z  |  E. y ( ph  /\  z  =  A ) } ) ) )
129, 11albid 1752 . . . . . 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 2914 . . . . 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 1718 . . . . . . . . 9  |-  ( (
ph  /\  z  =  A )  ->  E. y
( ph  /\  z  =  A ) )
1514ex 423 . . . . . . . 8  |-  ( ph  ->  ( z  =  A  ->  E. y ( ph  /\  z  =  A ) ) )
1615alrimiv 1617 . . . . . . 7  |-  ( ph  ->  A. z ( z  =  A  ->  E. y
( ph  /\  z  =  A ) ) )
17 intab.1 . . . . . . . 8  |-  A  e. 
_V
1817sbc6 3017 . . . . . . 7  |-  ( [. A  /  z ]. E. y ( ph  /\  z  =  A )  <->  A. z ( z  =  A  ->  E. y
( ph  /\  z  =  A ) ) )
1916, 18sylibr 203 . . . . . 6  |-  ( ph  ->  [. A  /  z ]. E. y ( ph  /\  z  =  A ) )
20 df-sbc 2992 . . . . . 6  |-  ( [. A  /  z ]. E. y ( ph  /\  z  =  A )  <->  A  e.  { z  |  E. y ( ph  /\  z  =  A ) } )
2119, 20sylib 188 . . . . 5  |-  ( ph  ->  A  e.  { z  |  E. y (
ph  /\  z  =  A ) } )
2213, 21mpgbir 1537 . . . 4  |-  { z  |  E. y (
ph  /\  z  =  A ) }  e.  { x  |  A. y
( ph  ->  A  e.  x ) }
23 intss1 3877 . . . 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 8 . . 3  |-  |^| { x  |  A. y ( ph  ->  A  e.  x ) }  C_  { z  |  E. y ( ph  /\  z  =  A ) }
25 19.29r 1584 . . . . . . . 8  |-  ( ( E. y ( ph  /\  z  =  A )  /\  A. y (
ph  ->  A  e.  x
) )  ->  E. y
( ( ph  /\  z  =  A )  /\  ( ph  ->  A  e.  x ) ) )
26 simplr 731 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  =  A )  /\  ( ph  ->  A  e.  x
) )  ->  z  =  A )
27 pm3.35 570 . . . . . . . . . . 11  |-  ( (
ph  /\  ( ph  ->  A  e.  x ) )  ->  A  e.  x )
2827adantlr 695 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  =  A )  /\  ( ph  ->  A  e.  x
) )  ->  A  e.  x )
2926, 28eqeltrd 2357 . . . . . . . . 9  |-  ( ( ( ph  /\  z  =  A )  /\  ( ph  ->  A  e.  x
) )  ->  z  e.  x )
3029exlimiv 1666 . . . . . . . 8  |-  ( E. y ( ( ph  /\  z  =  A )  /\  ( ph  ->  A  e.  x ) )  ->  z  e.  x
)
3125, 30syl 15 . . . . . . 7  |-  ( ( E. y ( ph  /\  z  =  A )  /\  A. y (
ph  ->  A  e.  x
) )  ->  z  e.  x )
3231ex 423 . . . . . 6  |-  ( E. y ( ph  /\  z  =  A )  ->  ( A. y (
ph  ->  A  e.  x
)  ->  z  e.  x ) )
3332alrimiv 1617 . . . . 5  |-  ( E. y ( ph  /\  z  =  A )  ->  A. x ( A. y ( ph  ->  A  e.  x )  -> 
z  e.  x ) )
34 vex 2791 . . . . . 6  |-  z  e. 
_V
3534elintab 3873 . . . . 5  |-  ( z  e.  |^| { x  | 
A. y ( ph  ->  A  e.  x ) }  <->  A. x ( A. y ( ph  ->  A  e.  x )  -> 
z  e.  x ) )
3633, 35sylibr 203 . . . 4  |-  ( E. y ( ph  /\  z  =  A )  ->  z  e.  |^| { x  |  A. y ( ph  ->  A  e.  x ) } )
3736abssi 3248 . . 3  |-  { z  |  E. y (
ph  /\  z  =  A ) }  C_  |^|
{ x  |  A. y ( ph  ->  A  e.  x ) }
3824, 37eqssi 3195 . 2  |-  |^| { x  |  A. y ( ph  ->  A  e.  x ) }  =  { z  |  E. y (
ph  /\  z  =  A ) }
3938, 4eqtri 2303 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 358   A.wal 1527   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269   _Vcvv 2788   [.wsbc 2991    C_ wss 3152   |^|cint 3862
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-sbc 2992  df-in 3159  df-ss 3166  df-int 3863
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