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

Theorem snjust 3645
Description: Soundness justification theorem for df-sn 3646. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
snjust  |-  { x  |  x  =  A }  =  { y  |  y  =  A }
Distinct variable groups:    x, A    y, A

Proof of Theorem snjust
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2289 . . 3  |-  ( x  =  z  ->  (
x  =  A  <->  z  =  A ) )
21cbvabv 2402 . 2  |-  { x  |  x  =  A }  =  { z  |  z  =  A }
3 eqeq1 2289 . . 3  |-  ( z  =  y  ->  (
z  =  A  <->  y  =  A ) )
43cbvabv 2402 . 2  |-  { z  |  z  =  A }  =  { y  |  y  =  A }
52, 4eqtri 2303 1  |-  { x  |  x  =  A }  =  { y  |  y  =  A }
Colors of variables: wff set class
Syntax hints:    = wceq 1623   {cab 2269
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276
  Copyright terms: Public domain W3C validator