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Theorem bj-sbab 31445
Description: Remove dependency on ax-13 2102 from sbab 2589 (note the absence of DV conditions among variables in the LHS). (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-sbab  |-  ( x  =  y  ->  A  =  { z  |  [
y  /  x ]
z  e.  A }
)
Distinct variable groups:    z, A    x, z    y, z
Allowed substitution hints:    A( x, y)

Proof of Theorem bj-sbab
StepHypRef Expression
1 sbequ12 2094 . 2  |-  ( x  =  y  ->  (
z  e.  A  <->  [ y  /  x ] z  e.  A ) )
21bj-abbi2dv 31441 1  |-  ( x  =  y  ->  A  =  { z  |  [
y  /  x ]
z  e.  A }
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1455   [wsb 1808    e. wcel 1898   {cab 2448
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-ext 2442
This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458
This theorem is referenced by: (None)
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