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Theorem snjust 3787
Description: Soundness justification theorem for df-sn 3788. (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 2418 . . 3  |-  ( x  =  z  ->  (
x  =  A  <->  z  =  A ) )
21cbvabv 2531 . 2  |-  { x  |  x  =  A }  =  { z  |  z  =  A }
3 eqeq1 2418 . . 3  |-  ( z  =  y  ->  (
z  =  A  <->  y  =  A ) )
43cbvabv 2531 . 2  |-  { z  |  z  =  A }  =  { y  |  y  =  A }
52, 4eqtri 2432 1  |-  { x  |  x  =  A }  =  { y  |  y  =  A }
Colors of variables: wff set class
Syntax hints:    = wceq 1649   {cab 2398
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405
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