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Theorem eqsb3 2537
Description: Substitution applied to an atomic wff (class version of equsb3 2178). (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 2536 . . 3  |-  ( [ w  /  y ] y  =  A  <->  w  =  A )
21sbbii 1665 . 2  |-  ( [ x  /  w ] [ w  /  y ] y  =  A  <->  [ x  /  w ] w  =  A
)
3 nfv 1629 . . 3  |-  F/ w  y  =  A
43sbco2 2161 . 2  |-  ( [ x  /  w ] [ w  /  y ] y  =  A  <->  [ x  /  y ] y  =  A )
5 eqsb3lem 2536 . 2  |-  ( [ x  /  w ]
w  =  A  <->  x  =  A )
62, 4, 53bitr3i 267 1  |-  ( [ x  /  y ] y  =  A  <->  x  =  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1652   [wsb 1658
This theorem is referenced by:  pm13.183  3076  eqsbc3  3200
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-cleq 2429
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