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Theorem intab 4109
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 6037 or abexssex 6038 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 2449 . . . . . . . . . 10  |-  ( z  =  x  ->  (
z  =  A  <->  x  =  A ) )
21anbi2d 686 . . . . . . . . 9  |-  ( z  =  x  ->  (
( ph  /\  z  =  A )  <->  ( ph  /\  x  =  A ) ) )
32exbidv 1638 . . . . . . . 8  |-  ( z  =  x  ->  ( E. y ( ph  /\  z  =  A )  <->  E. y ( ph  /\  x  =  A )
) )
43cbvabv 2562 . . . . . . 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 2513 . . . . . 6  |-  { z  |  E. y (
ph  /\  z  =  A ) }  e.  _V
7 nfe1 1750 . . . . . . . . 9  |-  F/ y E. y ( ph  /\  z  =  A )
87nfab 2583 . . . . . . . 8  |-  F/_ y { z  |  E. y ( ph  /\  z  =  A ) }
98nfeq2 2590 . . . . . . 7  |-  F/ y  x  =  { z  |  E. y (
ph  /\  z  =  A ) }
10 eleq2 2504 . . . . . . . 8  |-  ( x  =  { z  |  E. y ( ph  /\  z  =  A ) }  ->  ( A  e.  x  <->  A  e.  { z  |  E. y (
ph  /\  z  =  A ) } ) )
1110imbi2d 309 . . . . . . 7  |-  ( x  =  { z  |  E. y ( ph  /\  z  =  A ) }  ->  ( ( ph  ->  A  e.  x
)  <->  ( ph  ->  A  e.  { z  |  E. y ( ph  /\  z  =  A ) } ) ) )
129, 11albid 1791 . . . . . 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 3091 . . . . 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 1765 . . . . . . . . 9  |-  ( (
ph  /\  z  =  A )  ->  E. y
( ph  /\  z  =  A ) )
1514ex 425 . . . . . . . 8  |-  ( ph  ->  ( z  =  A  ->  E. y ( ph  /\  z  =  A ) ) )
1615alrimiv 1643 . . . . . . 7  |-  ( ph  ->  A. z ( z  =  A  ->  E. y
( ph  /\  z  =  A ) ) )
17 intab.1 . . . . . . . 8  |-  A  e. 
_V
1817sbc6 3196 . . . . . . 7  |-  ( [. A  /  z ]. E. y ( ph  /\  z  =  A )  <->  A. z ( z  =  A  ->  E. y
( ph  /\  z  =  A ) ) )
1916, 18sylibr 205 . . . . . 6  |-  ( ph  ->  [. A  /  z ]. E. y ( ph  /\  z  =  A ) )
20 df-sbc 3171 . . . . . 6  |-  ( [. A  /  z ]. E. y ( ph  /\  z  =  A )  <->  A  e.  { z  |  E. y ( ph  /\  z  =  A ) } )
2119, 20sylib 190 . . . . 5  |-  ( ph  ->  A  e.  { z  |  E. y (
ph  /\  z  =  A ) } )
2213, 21mpgbir 1560 . . . 4  |-  { z  |  E. y (
ph  /\  z  =  A ) }  e.  { x  |  A. y
( ph  ->  A  e.  x ) }
23 intss1 4094 . . . 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 1609 . . . . . . . 8  |-  ( ( E. y ( ph  /\  z  =  A )  /\  A. y (
ph  ->  A  e.  x
) )  ->  E. y
( ( ph  /\  z  =  A )  /\  ( ph  ->  A  e.  x ) ) )
26 simplr 733 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  =  A )  /\  ( ph  ->  A  e.  x
) )  ->  z  =  A )
27 pm3.35 572 . . . . . . . . . . 11  |-  ( (
ph  /\  ( ph  ->  A  e.  x ) )  ->  A  e.  x )
2827adantlr 697 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  =  A )  /\  ( ph  ->  A  e.  x
) )  ->  A  e.  x )
2926, 28eqeltrd 2517 . . . . . . . . 9  |-  ( ( ( ph  /\  z  =  A )  /\  ( ph  ->  A  e.  x
) )  ->  z  e.  x )
3029exlimiv 1646 . . . . . . . 8  |-  ( E. y ( ( ph  /\  z  =  A )  /\  ( ph  ->  A  e.  x ) )  ->  z  e.  x
)
3125, 30syl 16 . . . . . . 7  |-  ( ( E. y ( ph  /\  z  =  A )  /\  A. y (
ph  ->  A  e.  x
) )  ->  z  e.  x )
3231ex 425 . . . . . 6  |-  ( E. y ( ph  /\  z  =  A )  ->  ( A. y (
ph  ->  A  e.  x
)  ->  z  e.  x ) )
3332alrimiv 1643 . . . . 5  |-  ( E. y ( ph  /\  z  =  A )  ->  A. x ( A. y ( ph  ->  A  e.  x )  -> 
z  e.  x ) )
34 vex 2968 . . . . . 6  |-  z  e. 
_V
3534elintab 4090 . . . . 5  |-  ( z  e.  |^| { x  | 
A. y ( ph  ->  A  e.  x ) }  <->  A. x ( A. y ( ph  ->  A  e.  x )  -> 
z  e.  x ) )
3633, 35sylibr 205 . . . 4  |-  ( E. y ( ph  /\  z  =  A )  ->  z  e.  |^| { x  |  A. y ( ph  ->  A  e.  x ) } )
3736abssi 3407 . . 3  |-  { z  |  E. y (
ph  /\  z  =  A ) }  C_  |^|
{ x  |  A. y ( ph  ->  A  e.  x ) }
3824, 37eqssi 3353 . 2  |-  |^| { x  |  A. y ( ph  ->  A  e.  x ) }  =  { z  |  E. y (
ph  /\  z  =  A ) }
3938, 4eqtri 2463 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 360   A.wal 1550   E.wex 1551    = wceq 1654    e. wcel 1728   {cab 2429   _Vcvv 2965   [.wsbc 3170    C_ wss 3309   |^|cint 4079
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424
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 1661  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-v 2967  df-sbc 3171  df-in 3316  df-ss 3323  df-int 4080
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