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Theorem dfrab3ss 3459
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 3179 . . 3  |-  ( A 
C_  B  <->  ( A  i^i  B )  =  A )
2 ineq1 3376 . . . 4  |-  ( ( A  i^i  B )  =  A  ->  (
( A  i^i  B
)  i^i  { x  |  ph } )  =  ( A  i^i  {
x  |  ph }
) )
32eqcomd 2301 . . 3  |-  ( ( A  i^i  B )  =  A  ->  ( A  i^i  { x  | 
ph } )  =  ( ( A  i^i  B )  i^i  { x  |  ph } ) )
41, 3sylbi 187 . 2  |-  ( A 
C_  B  ->  ( A  i^i  { x  | 
ph } )  =  ( ( A  i^i  B )  i^i  { x  |  ph } ) )
5 dfrab3 3457 . 2  |-  { x  e.  A  |  ph }  =  ( A  i^i  { x  |  ph }
)
6 dfrab3 3457 . . . 4  |-  { x  e.  B  |  ph }  =  ( B  i^i  { x  |  ph }
)
76ineq2i 3380 . . 3  |-  ( A  i^i  { x  e.  B  |  ph }
)  =  ( A  i^i  ( B  i^i  { x  |  ph }
) )
8 inass 3392 . . 3  |-  ( ( A  i^i  B )  i^i  { x  | 
ph } )  =  ( A  i^i  ( B  i^i  { x  | 
ph } ) )
97, 8eqtr4i 2319 . 2  |-  ( A  i^i  { x  e.  B  |  ph }
)  =  ( ( A  i^i  B )  i^i  { x  | 
ph } )
104, 5, 93eqtr4g 2353 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 1632   {cab 2282   {crab 2560    i^i cin 3164    C_ wss 3165
This theorem is referenced by:  proot1hash  27622
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rab 2565  df-v 2803  df-in 3172  df-ss 3179
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