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Theorem eqsb3 2587
Description: Substitution applied to an atomic wff (class version of equsb3 2159). (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 2586 . . 3  |-  ( [ w  /  y ] y  =  A  <->  w  =  A )
21sbbii 1718 . 2  |-  ( [ x  /  w ] [ w  /  y ] y  =  A  <->  [ x  /  w ] w  =  A
)
3 nfv 1683 . . 3  |-  F/ w  y  =  A
43sbco2 2134 . 2  |-  ( [ x  /  w ] [ w  /  y ] y  =  A  <->  [ x  /  y ] y  =  A )
5 eqsb3lem 2586 . 2  |-  ( [ x  /  w ]
w  =  A  <->  x  =  A )
62, 4, 53bitr3i 275 1  |-  ( [ x  /  y ] y  =  A  <->  x  =  A )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1379   [wsb 1711
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1597  df-nf 1600  df-sb 1712  df-cleq 2459
This theorem is referenced by:  pm13.183  3249  eqsbc3  3376  frege55lem1b  37288
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