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Theorem dfrab2 3456
Description: Alternate definition of restricted class abstraction. (Contributed by NM, 20-Sep-2003.)
Assertion
Ref Expression
dfrab2  |-  { x  e.  A  |  ph }  =  ( { x  |  ph }  i^i  A
)
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem dfrab2
StepHypRef Expression
1 df-rab 2565 . 2  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
2 inab 3449 . . 3  |-  ( { x  |  x  e.  A }  i^i  {
x  |  ph }
)  =  { x  |  ( x  e.  A  /\  ph ) }
3 abid2 2413 . . . 4  |-  { x  |  x  e.  A }  =  A
43ineq1i 3379 . . 3  |-  ( { x  |  x  e.  A }  i^i  {
x  |  ph }
)  =  ( A  i^i  { x  | 
ph } )
52, 4eqtr3i 2318 . 2  |-  { x  |  ( x  e.  A  /\  ph ) }  =  ( A  i^i  { x  |  ph } )
6 incom 3374 . 2  |-  ( A  i^i  { x  | 
ph } )  =  ( { x  | 
ph }  i^i  A
)
71, 5, 63eqtri 2320 1  |-  { x  e.  A  |  ph }  =  ( { x  |  ph }  i^i  A
)
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1632    e. wcel 1696   {cab 2282   {crab 2560    i^i cin 3164
This theorem is referenced by:  epse  4392  dfsup2OLD  7212  psrbagsn  16252  ismbl  18901  orvcval4  23676  fvline2  24841
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
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