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Theorem rabrab 27172
Description: Abstract builder restricted to another restricted abstract builder (Contributed by Thierry Arnoux, 30-Aug-2017.)
Assertion
Ref Expression
rabrab  |-  { x  e.  { x  e.  A  |  ph }  |  ps }  =  { x  e.  A  |  ( ph  /\  ps ) }

Proof of Theorem rabrab
StepHypRef Expression
1 rabid 3038 . . . . 5  |-  ( x  e.  { x  e.  A  |  ph }  <->  ( x  e.  A  /\  ph ) )
21anbi1i 695 . . . 4  |-  ( ( x  e.  { x  e.  A  |  ph }  /\  ps )  <->  ( (
x  e.  A  /\  ph )  /\  ps )
)
3 anass 649 . . . 4  |-  ( ( ( x  e.  A  /\  ph )  /\  ps ) 
<->  ( x  e.  A  /\  ( ph  /\  ps ) ) )
42, 3bitri 249 . . 3  |-  ( ( x  e.  { x  e.  A  |  ph }  /\  ps )  <->  ( x  e.  A  /\  ( ph  /\  ps ) ) )
54abbii 2601 . 2  |-  { x  |  ( x  e. 
{ x  e.  A  |  ph }  /\  ps ) }  =  {
x  |  ( x  e.  A  /\  ( ph  /\  ps ) ) }
6 df-rab 2823 . 2  |-  { x  e.  { x  e.  A  |  ph }  |  ps }  =  { x  |  ( x  e. 
{ x  e.  A  |  ph }  /\  ps ) }
7 df-rab 2823 . 2  |-  { x  e.  A  |  ( ph  /\  ps ) }  =  { x  |  ( x  e.  A  /\  ( ph  /\  ps ) ) }
85, 6, 73eqtr4i 2506 1  |-  { x  e.  { x  e.  A  |  ph }  |  ps }  =  { x  e.  A  |  ( ph  /\  ps ) }
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1379    e. wcel 1767   {cab 2452   {crab 2818
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-rab 2823
This theorem is referenced by:  fpwrelmapffs  27326
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