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Theorem snjust 3721
Description: Soundness justification theorem for df-sn 3722. (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 2364 . . 3  |-  ( x  =  z  ->  (
x  =  A  <->  z  =  A ) )
21cbvabv 2477 . 2  |-  { x  |  x  =  A }  =  { z  |  z  =  A }
3 eqeq1 2364 . . 3  |-  ( z  =  y  ->  (
z  =  A  <->  y  =  A ) )
43cbvabv 2477 . 2  |-  { z  |  z  =  A }  =  { y  |  y  =  A }
52, 4eqtri 2378 1  |-  { x  |  x  =  A }  =  { y  |  y  =  A }
Colors of variables: wff set class
Syntax hints:    = wceq 1642   {cab 2344
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351
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