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Theorem pwjust 3964
Description: Soundness justification theorem for df-pw 3965. (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 3465 . . 3  |-  ( x  =  z  ->  (
x  C_  A  <->  z  C_  A ) )
21cbvabv 2586 . 2  |-  { x  |  x  C_  A }  =  { z  |  z 
C_  A }
3 sseq1 3465 . . 3  |-  ( z  =  y  ->  (
z  C_  A  <->  y  C_  A ) )
43cbvabv 2586 . 2  |-  { z  |  z  C_  A }  =  { y  |  y  C_  A }
52, 4eqtri 2484 1  |-  { x  |  x  C_  A }  =  { y  |  y 
C_  A }
Colors of variables: wff setvar class
Syntax hints:    = wceq 1455   {cab 2448    C_ wss 3416
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-in 3423  df-ss 3430
This theorem is referenced by: (None)
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