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Theorem pwjust 3792
Description: Soundness justification theorem for df-pw 3793. (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 3361 . . 3  |-  ( x  =  z  ->  (
x  C_  A  <->  z  C_  A ) )
21cbvabv 2554 . 2  |-  { x  |  x  C_  A }  =  { z  |  z 
C_  A }
3 sseq1 3361 . . 3  |-  ( z  =  y  ->  (
z  C_  A  <->  y  C_  A ) )
43cbvabv 2554 . 2  |-  { z  |  z  C_  A }  =  { y  |  y  C_  A }
52, 4eqtri 2455 1  |-  { x  |  x  C_  A }  =  { y  |  y 
C_  A }
Colors of variables: wff set class
Syntax hints:    = wceq 1652   {cab 2421    C_ wss 3312
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-in 3319  df-ss 3326
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