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Theorem clelsb1f 2333
Description: Substitution for the first argument of the membership predicate in an atomic formula (class version of elsb1 2165). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (Revised by Thierry Arnoux, 13-Mar-2017.)
Hypothesis
Ref Expression
clelsb1f.1  |-  F/_ x A
Assertion
Ref Expression
clelsb1f  |-  ( [ y  /  x ]
x  e.  A  <->  y  e.  A )

Proof of Theorem clelsb1f
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 clelsb1f.1 . . . 4  |-  F/_ x A
21nfcri 2323 . . 3  |-  F/ x  w  e.  A
32sbco2 1975 . 2  |-  ( [ y  /  x ] [ x  /  w ] w  e.  A  <->  [ y  /  w ]
w  e.  A )
4 nfv 1538 . . . 4  |-  F/ w  x  e.  A
5 eleq1w 2248 . . . 4  |-  ( w  =  x  ->  (
w  e.  A  <->  x  e.  A ) )
64, 5sbie 1801 . . 3  |-  ( [ x  /  w ]
w  e.  A  <->  x  e.  A )
76sbbii 1775 . 2  |-  ( [ y  /  x ] [ x  /  w ] w  e.  A  <->  [ y  /  x ]
x  e.  A )
8 nfv 1538 . . 3  |-  F/ w  y  e.  A
9 eleq1w 2248 . . 3  |-  ( w  =  y  ->  (
w  e.  A  <->  y  e.  A ) )
108, 9sbie 1801 . 2  |-  ( [ y  /  w ]
w  e.  A  <->  y  e.  A )
113, 7, 103bitr3i 210 1  |-  ( [ y  /  x ]
x  e.  A  <->  y  e.  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 105   [wsb 1772    e. wcel 2158   F/_wnfc 2316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773  df-cleq 2180  df-clel 2183  df-nfc 2318
This theorem is referenced by:  rmo3f  2946
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