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Theorem rabss3d 26016
Description: Subclass law for restricted abstraction. (Contributed by Thierry Arnoux, 25-Sep-2017.)
Hypothesis
Ref Expression
rabss3d.1  |-  ( (
ph  /\  ( x  e.  A  /\  ps )
)  ->  x  e.  B )
Assertion
Ref Expression
rabss3d  |-  ( ph  ->  { x  e.  A  |  ps }  C_  { x  e.  B  |  ps } )
Distinct variable groups:    x, A    x, B    ph, x
Allowed substitution hint:    ps( x)

Proof of Theorem rabss3d
StepHypRef Expression
1 nfv 1674 . 2  |-  F/ x ph
2 nfrab1 2983 . 2  |-  F/_ x { x  e.  A  |  ps }
3 nfrab1 2983 . 2  |-  F/_ x { x  e.  B  |  ps }
4 rabss3d.1 . . . . 5  |-  ( (
ph  /\  ( x  e.  A  /\  ps )
)  ->  x  e.  B )
5 simprr 756 . . . . 5  |-  ( (
ph  /\  ( x  e.  A  /\  ps )
)  ->  ps )
64, 5jca 532 . . . 4  |-  ( (
ph  /\  ( x  e.  A  /\  ps )
)  ->  ( x  e.  B  /\  ps )
)
76ex 434 . . 3  |-  ( ph  ->  ( ( x  e.  A  /\  ps )  ->  ( x  e.  B  /\  ps ) ) )
8 rabid 2979 . . 3  |-  ( x  e.  { x  e.  A  |  ps }  <->  ( x  e.  A  /\  ps ) )
9 rabid 2979 . . 3  |-  ( x  e.  { x  e.  B  |  ps }  <->  ( x  e.  B  /\  ps ) )
107, 8, 93imtr4g 270 . 2  |-  ( ph  ->  ( x  e.  {
x  e.  A  |  ps }  ->  x  e.  { x  e.  B  |  ps } ) )
111, 2, 3, 10ssrd 3445 1  |-  ( ph  ->  { x  e.  A  |  ps }  C_  { x  e.  B  |  ps } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1757   {crab 2796    C_ wss 3412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-rab 2801  df-in 3419  df-ss 3426
This theorem is referenced by:  xpinpreima2  26457
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