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Theorem dfrab3ss 3757
Description: Restricted class abstraction with a common superset. (Contributed by Stefan O'Rear, 12-Sep-2015.) (Proof shortened by Mario Carneiro, 8-Nov-2015.)
Assertion
Ref Expression
dfrab3ss  |-  ( A 
C_  B  ->  { x  e.  A  |  ph }  =  ( A  i^i  { x  e.  B  |  ph } ) )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    ph( x)

Proof of Theorem dfrab3ss
StepHypRef Expression
1 df-ss 3456 . . 3  |-  ( A 
C_  B  <->  ( A  i^i  B )  =  A )
2 ineq1 3663 . . . 4  |-  ( ( A  i^i  B )  =  A  ->  (
( A  i^i  B
)  i^i  { x  |  ph } )  =  ( A  i^i  {
x  |  ph }
) )
32eqcomd 2437 . . 3  |-  ( ( A  i^i  B )  =  A  ->  ( A  i^i  { x  | 
ph } )  =  ( ( A  i^i  B )  i^i  { x  |  ph } ) )
41, 3sylbi 198 . 2  |-  ( A 
C_  B  ->  ( A  i^i  { x  | 
ph } )  =  ( ( A  i^i  B )  i^i  { x  |  ph } ) )
5 dfrab3 3754 . 2  |-  { x  e.  A  |  ph }  =  ( A  i^i  { x  |  ph }
)
6 dfrab3 3754 . . . 4  |-  { x  e.  B  |  ph }  =  ( B  i^i  { x  |  ph }
)
76ineq2i 3667 . . 3  |-  ( A  i^i  { x  e.  B  |  ph }
)  =  ( A  i^i  ( B  i^i  { x  |  ph }
) )
8 inass 3678 . . 3  |-  ( ( A  i^i  B )  i^i  { x  | 
ph } )  =  ( A  i^i  ( B  i^i  { x  | 
ph } ) )
97, 8eqtr4i 2461 . 2  |-  ( A  i^i  { x  e.  B  |  ph }
)  =  ( ( A  i^i  B )  i^i  { x  | 
ph } )
104, 5, 93eqtr4g 2495 1  |-  ( A 
C_  B  ->  { x  e.  A  |  ph }  =  ( A  i^i  { x  e.  B  |  ph } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437   {cab 2414   {crab 2786    i^i cin 3441    C_ wss 3442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-rab 2791  df-v 3089  df-in 3449  df-ss 3456
This theorem is referenced by:  cusgrasizeindslem2  25047  mbfposadd  31692  proot1hash  35776
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