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Theorem snjust 3586
Description: Soundness justification theorem for df-sn 3587. (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 2177 . . 3  |-  ( x  =  z  ->  (
x  =  A  <->  z  =  A ) )
21cbvabv 2295 . 2  |-  { x  |  x  =  A }  =  { z  |  z  =  A }
3 eqeq1 2177 . . 3  |-  ( z  =  y  ->  (
z  =  A  <->  y  =  A ) )
43cbvabv 2295 . 2  |-  { z  |  z  =  A }  =  { y  |  y  =  A }
52, 4eqtri 2191 1  |-  { x  |  x  =  A }  =  { y  |  y  =  A }
Colors of variables: wff set class
Syntax hints:    = wceq 1348   {cab 2156
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163
This theorem is referenced by: (None)
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