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Theorem rabss3d 27614
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 1712 . 2  |-  F/ x ph
2 nfrab1 3035 . 2  |-  F/_ x { x  e.  A  |  ps }
3 nfrab1 3035 . 2  |-  F/_ x { x  e.  B  |  ps }
4 rabss3d.1 . . . . 5  |-  ( (
ph  /\  ( x  e.  A  /\  ps )
)  ->  x  e.  B )
5 simprr 755 . . . . 5  |-  ( (
ph  /\  ( x  e.  A  /\  ps )
)  ->  ps )
64, 5jca 530 . . . 4  |-  ( (
ph  /\  ( x  e.  A  /\  ps )
)  ->  ( x  e.  B  /\  ps )
)
76ex 432 . . 3  |-  ( ph  ->  ( ( x  e.  A  /\  ps )  ->  ( x  e.  B  /\  ps ) ) )
8 rabid 3031 . . 3  |-  ( x  e.  { x  e.  A  |  ps }  <->  ( x  e.  A  /\  ps ) )
9 rabid 3031 . . 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 3494 1  |-  ( ph  ->  { x  e.  A  |  ps }  C_  { x  e.  B  |  ps } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    e. wcel 1823   {crab 2808    C_ wss 3461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rab 2813  df-in 3468  df-ss 3475
This theorem is referenced by:  xpinpreima2  28127
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