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Theorem eqsb3 2384
Description: Substitution applied to an atomic wff (class version of equsb3 2041). (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 2383 . . 3  |-  ( [ w  /  y ] y  =  A  <->  w  =  A )
21sbbii 1634 . 2  |-  ( [ x  /  w ] [ w  /  y ] y  =  A  <->  [ x  /  w ] w  =  A
)
3 nfv 1605 . . 3  |-  F/ w  y  =  A
43sbco2 2026 . 2  |-  ( [ x  /  w ] [ w  /  y ] y  =  A  <->  [ x  /  y ] y  =  A )
5 eqsb3lem 2383 . 2  |-  ( [ x  /  w ]
w  =  A  <->  x  =  A )
62, 4, 53bitr3i 266 1  |-  ( [ x  /  y ] y  =  A  <->  x  =  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1623   [wsb 1629
This theorem is referenced by:  pm13.183  2908  eqsbc3  3030  sb8iota  5226
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-cleq 2276
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