MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pwjust Unicode version

Theorem pwjust 3744
Description: Soundness justification theorem for df-pw 3745. (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 3313 . . 3  |-  ( x  =  z  ->  (
x  C_  A  <->  z  C_  A ) )
21cbvabv 2507 . 2  |-  { x  |  x  C_  A }  =  { z  |  z 
C_  A }
3 sseq1 3313 . . 3  |-  ( z  =  y  ->  (
z  C_  A  <->  y  C_  A ) )
43cbvabv 2507 . 2  |-  { z  |  z  C_  A }  =  { y  |  y  C_  A }
52, 4eqtri 2408 1  |-  { x  |  x  C_  A }  =  { y  |  y 
C_  A }
Colors of variables: wff set class
Syntax hints:    = wceq 1649   {cab 2374    C_ wss 3264
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-in 3271  df-ss 3278
  Copyright terms: Public domain W3C validator