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Theorem pwjust 3986
Description: Soundness justification theorem for df-pw 3987. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
pwjust  |-  { x  |  x  C_  A }  =  { y  |  y 
C_  A }
Distinct variable groups:    x, A    y, A

Proof of Theorem pwjust
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 sseq1 3491 . . 3  |-  ( x  =  z  ->  (
x  C_  A  <->  z  C_  A ) )
21cbvabv 2572 . 2  |-  { x  |  x  C_  A }  =  { z  |  z 
C_  A }
3 sseq1 3491 . . 3  |-  ( z  =  y  ->  (
z  C_  A  <->  y  C_  A ) )
43cbvabv 2572 . 2  |-  { z  |  z  C_  A }  =  { y  |  y  C_  A }
52, 4eqtri 2458 1  |-  { x  |  x  C_  A }  =  { y  |  y 
C_  A }
Colors of variables: wff setvar class
Syntax hints:    = wceq 1437   {cab 2414    C_ wss 3442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-in 3449  df-ss 3456
This theorem is referenced by: (None)
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