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Theorem bj-rababwv 31007
Description: A weak version of rabab 3077 not using df-clel 2397 nor df-v 3061 (but requiring ax-ext 2380). A version without dv condition is provable by replacing bj-vexwv 30997 with bj-vexw 30995 in the proof, hence requiring ax-13 2026. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
bj-rababwv.1  |-  ps
Assertion
Ref Expression
bj-rababwv  |-  { x  e.  { y  |  ps }  |  ph }  =  { x  |  ph }
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    ps( x, y)

Proof of Theorem bj-rababwv
StepHypRef Expression
1 df-rab 2763 . 2  |-  { x  e.  { y  |  ps }  |  ph }  =  { x  |  (
x  e.  { y  |  ps }  /\  ph ) }
2 bj-rababwv.1 . . . . 5  |-  ps
32bj-vexwv 30997 . . . 4  |-  x  e. 
{ y  |  ps }
43biantrur 504 . . 3  |-  ( ph  <->  ( x  e.  { y  |  ps }  /\  ph ) )
54bj-abbii 30927 . 2  |-  { x  |  ph }  =  {
x  |  ( x  e.  { y  |  ps }  /\  ph ) }
61, 5eqtr4i 2434 1  |-  { x  e.  { y  |  ps }  |  ph }  =  { x  |  ph }
Colors of variables: wff setvar class
Syntax hints:    /\ wa 367    = wceq 1405    e. wcel 1842   {cab 2387   {crab 2758
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-rab 2763
This theorem is referenced by: (None)
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