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Theorem intabssd 36260
Description: When for each element  y there is a subset  A which may substituted for  x such that  y satisfying  ch implies  x satisfies  ps then the intersection of all  x that satisfy  ps is a subclass the intersection of all  y that satisfy  ch. (Contributed by RP, 17-Oct-2020.)
Hypotheses
Ref Expression
intabssd.ex  |-  ( ph  ->  A  e.  V )
intabssd.sub  |-  ( (
ph  /\  x  =  A )  ->  ( ch  ->  ps ) )
intabssd.ss  |-  ( ph  ->  A  C_  y )
Assertion
Ref Expression
intabssd  |-  ( ph  ->  |^| { x  |  ps }  C_  |^| { y  |  ch } )
Distinct variable groups:    ch, x    ps, y    x, y, ph    x, A
Allowed substitution hints:    ps( x)    ch( y)    A( y)    V( x, y)

Proof of Theorem intabssd
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 intabssd.ex . . . . 5  |-  ( ph  ->  A  e.  V )
2 intabssd.sub . . . . . 6  |-  ( (
ph  /\  x  =  A )  ->  ( ch  ->  ps ) )
3 eleq2 2528 . . . . . . . 8  |-  ( x  =  A  ->  (
z  e.  x  <->  z  e.  A ) )
43biimpd 212 . . . . . . 7  |-  ( x  =  A  ->  (
z  e.  x  -> 
z  e.  A ) )
5 intabssd.ss . . . . . . . 8  |-  ( ph  ->  A  C_  y )
65sseld 3442 . . . . . . 7  |-  ( ph  ->  ( z  e.  A  ->  z  e.  y ) )
74, 6sylan9r 668 . . . . . 6  |-  ( (
ph  /\  x  =  A )  ->  (
z  e.  x  -> 
z  e.  y ) )
82, 7imim12d 77 . . . . 5  |-  ( (
ph  /\  x  =  A )  ->  (
( ps  ->  z  e.  x )  ->  ( ch  ->  z  e.  y ) ) )
91, 8spcimdv 3142 . . . 4  |-  ( ph  ->  ( A. x ( ps  ->  z  e.  x )  ->  ( ch  ->  z  e.  y ) ) )
109alrimdv 1785 . . 3  |-  ( ph  ->  ( A. x ( ps  ->  z  e.  x )  ->  A. y
( ch  ->  z  e.  y ) ) )
11 vex 3059 . . . 4  |-  z  e. 
_V
1211elintab 4258 . . 3  |-  ( z  e.  |^| { x  |  ps }  <->  A. x
( ps  ->  z  e.  x ) )
1311elintab 4258 . . 3  |-  ( z  e.  |^| { y  |  ch }  <->  A. y
( ch  ->  z  e.  y ) )
1410, 12, 133imtr4g 278 . 2  |-  ( ph  ->  ( z  e.  |^| { x  |  ps }  ->  z  e.  |^| { y  |  ch } ) )
1514ssrdv 3449 1  |-  ( ph  ->  |^| { x  |  ps }  C_  |^| { y  |  ch } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 375   A.wal 1452    = wceq 1454    e. wcel 1897   {cab 2447    C_ wss 3415   |^|cint 4247
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441
This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-v 3058  df-in 3422  df-ss 3429  df-int 4248
This theorem is referenced by:  clcnvlem  36274
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