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Theorem eqsb3 2183
Description: Substitution applied to an atomic wff (class version of equsb3 1867). (Contributed by Rodolfo Medina, 28-Apr-2010.)
Assertion
Ref Expression
eqsb3  |-  ( [ x  /  y ] y  =  A  <->  x  =  A )
Distinct variable group:    y, A
Allowed substitution hint:    A( x)

Proof of Theorem eqsb3
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 eqsb3lem 2182 . . 3  |-  ( [ w  /  y ] y  =  A  <->  w  =  A )
21sbbii 1689 . 2  |-  ( [ x  /  w ] [ w  /  y ] y  =  A  <->  [ x  /  w ] w  =  A
)
3 nfv 1462 . . 3  |-  F/ w  y  =  A
43sbco2 1881 . 2  |-  ( [ x  /  w ] [ w  /  y ] y  =  A  <->  [ x  /  y ] y  =  A )
5 eqsb3lem 2182 . 2  |-  ( [ x  /  w ]
w  =  A  <->  x  =  A )
62, 4, 53bitr3i 208 1  |-  ( [ x  /  y ] y  =  A  <->  x  =  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 103    = wceq 1285   [wsb 1686
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-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-cleq 2075
This theorem is referenced by:  pm13.183  2733  eqsbc3  2854
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