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Theorem snjust 3821
Description: Soundness justification theorem for df-sn 3822. (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 2444 . . 3  |-  ( x  =  z  ->  (
x  =  A  <->  z  =  A ) )
21cbvabv 2557 . 2  |-  { x  |  x  =  A }  =  { z  |  z  =  A }
3 eqeq1 2444 . . 3  |-  ( z  =  y  ->  (
z  =  A  <->  y  =  A ) )
43cbvabv 2557 . 2  |-  { z  |  z  =  A }  =  { y  |  y  =  A }
52, 4eqtri 2458 1  |-  { x  |  x  =  A }  =  { y  |  y  =  A }
Colors of variables: wff set class
Syntax hints:    = wceq 1653   {cab 2424
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431
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