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Theorem clelsb1 2275
Description: Substitution for the first argument of the membership predicate in an atomic formula (class version of elsb1 2148). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
Assertion
Ref Expression
clelsb1  |-  ( [ y  /  x ]
x  e.  A  <->  y  e.  A )
Distinct variable group:    x, A
Allowed substitution hint:    A( y)

Proof of Theorem clelsb1
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 nfv 1521 . . 3  |-  F/ x  w  e.  A
21sbco2 1958 . 2  |-  ( [ y  /  x ] [ x  /  w ] w  e.  A  <->  [ y  /  w ]
w  e.  A )
3 nfv 1521 . . . 4  |-  F/ w  x  e.  A
4 eleq1 2233 . . . 4  |-  ( w  =  x  ->  (
w  e.  A  <->  x  e.  A ) )
53, 4sbie 1784 . . 3  |-  ( [ x  /  w ]
w  e.  A  <->  x  e.  A )
65sbbii 1758 . 2  |-  ( [ y  /  x ] [ x  /  w ] w  e.  A  <->  [ y  /  x ]
x  e.  A )
7 nfv 1521 . . 3  |-  F/ w  y  e.  A
8 eleq1 2233 . . 3  |-  ( w  =  y  ->  (
w  e.  A  <->  y  e.  A ) )
97, 8sbie 1784 . 2  |-  ( [ y  /  w ]
w  e.  A  <->  y  e.  A )
102, 6, 93bitr3i 209 1  |-  ( [ y  /  x ]
x  e.  A  <->  y  e.  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 104   [wsb 1755    e. wcel 2141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-cleq 2163  df-clel 2166
This theorem is referenced by:  hblem  2278  nfraldya  2505  nfrexdya  2506  cbvreu  2694  sbcel1v  3017  rmo3  3046  setindel  4520  elirr  4523  en2lp  4536  zfregfr  4556  tfi  4564  bdcriota  13840
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