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Theorem vjust 2620
Description: Soundness justification theorem for df-v 2621. (Contributed by Rodolfo Medina, 27-Apr-2010.)
Assertion
Ref Expression
vjust  |-  { x  |  x  =  x }  =  { y  |  y  =  y }

Proof of Theorem vjust
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 equid 1634 . . . . 5  |-  x  =  x
21sbt 1714 . . . 4  |-  [ z  /  x ] x  =  x
3 equid 1634 . . . . 5  |-  y  =  y
43sbt 1714 . . . 4  |-  [ z  /  y ] y  =  y
52, 42th 172 . . 3  |-  ( [ z  /  x ]
x  =  x  <->  [ z  /  y ] y  =  y )
6 df-clab 2075 . . 3  |-  ( z  e.  { x  |  x  =  x }  <->  [ z  /  x ]
x  =  x )
7 df-clab 2075 . . 3  |-  ( z  e.  { y  |  y  =  y }  <->  [ z  /  y ] y  =  y )
85, 6, 73bitr4i 210 . 2  |-  ( z  e.  { x  |  x  =  x }  <->  z  e.  { y  |  y  =  y } )
98eqriv 2085 1  |-  { x  |  x  =  x }  =  { y  |  y  =  y }
Colors of variables: wff set class
Syntax hints:    = wceq 1289    e. wcel 1438   [wsb 1692   {cab 2074
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081
This theorem is referenced by: (None)
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