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Theorem dfrab3ss 3354
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 3084 . . 3  |-  ( A 
C_  B  <->  ( A  i^i  B )  =  A )
2 ineq1 3270 . . . 4  |-  ( ( A  i^i  B )  =  A  ->  (
( A  i^i  B
)  i^i  { x  |  ph } )  =  ( A  i^i  {
x  |  ph }
) )
32eqcomd 2145 . . 3  |-  ( ( A  i^i  B )  =  A  ->  ( A  i^i  { x  | 
ph } )  =  ( ( A  i^i  B )  i^i  { x  |  ph } ) )
41, 3sylbi 120 . 2  |-  ( A 
C_  B  ->  ( A  i^i  { x  | 
ph } )  =  ( ( A  i^i  B )  i^i  { x  |  ph } ) )
5 dfrab3 3352 . 2  |-  { x  e.  A  |  ph }  =  ( A  i^i  { x  |  ph }
)
6 dfrab3 3352 . . . 4  |-  { x  e.  B  |  ph }  =  ( B  i^i  { x  |  ph }
)
76ineq2i 3274 . . 3  |-  ( A  i^i  { x  e.  B  |  ph }
)  =  ( A  i^i  ( B  i^i  { x  |  ph }
) )
8 inass 3286 . . 3  |-  ( ( A  i^i  B )  i^i  { x  | 
ph } )  =  ( A  i^i  ( B  i^i  { x  | 
ph } ) )
97, 8eqtr4i 2163 . 2  |-  ( A  i^i  { x  e.  B  |  ph }
)  =  ( ( A  i^i  B )  i^i  { x  | 
ph } )
104, 5, 93eqtr4g 2197 1  |-  ( A 
C_  B  ->  { x  e.  A  |  ph }  =  ( A  i^i  { x  e.  B  |  ph } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331   {cab 2125   {crab 2420    i^i cin 3070    C_ wss 3071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rab 2425  df-v 2688  df-in 3077  df-ss 3084
This theorem is referenced by: (None)
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