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

Theorem pwjust 4016
Description: Soundness justification theorem for df-pw 4017. (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 3520 . . 3  |-  ( x  =  z  ->  (
x  C_  A  <->  z  C_  A ) )
21cbvabv 2600 . 2  |-  { x  |  x  C_  A }  =  { z  |  z 
C_  A }
3 sseq1 3520 . . 3  |-  ( z  =  y  ->  (
z  C_  A  <->  y  C_  A ) )
43cbvabv 2600 . 2  |-  { z  |  z  C_  A }  =  { y  |  y  C_  A }
52, 4eqtri 2486 1  |-  { x  |  x  C_  A }  =  { y  |  y 
C_  A }
Colors of variables: wff setvar class
Syntax hints:    = wceq 1395   {cab 2442    C_ wss 3471
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-in 3478  df-ss 3485
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator