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Theorem eleq2w 2149
Description: Weaker version of eleq2 2151 (but more general than elequ2 1648) not depending on ax-ext 2070 nor df-cleq 2081. (Contributed by BJ, 29-Sep-2019.)
Assertion
Ref Expression
eleq2w  |-  ( x  =  y  ->  ( A  e.  x  <->  A  e.  y ) )

Proof of Theorem eleq2w
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 elequ2 1648 . . . 4  |-  ( x  =  y  ->  (
z  e.  x  <->  z  e.  y ) )
21anbi2d 452 . . 3  |-  ( x  =  y  ->  (
( z  =  A  /\  z  e.  x
)  <->  ( z  =  A  /\  z  e.  y ) ) )
32exbidv 1753 . 2  |-  ( x  =  y  ->  ( E. z ( z  =  A  /\  z  e.  x )  <->  E. z
( z  =  A  /\  z  e.  y ) ) )
4 df-clel 2084 . 2  |-  ( A  e.  x  <->  E. z
( z  =  A  /\  z  e.  x
) )
5 df-clel 2084 . 2  |-  ( A  e.  y  <->  E. z
( z  =  A  /\  z  e.  y ) )
63, 4, 53bitr4g 221 1  |-  ( x  =  y  ->  ( A  e.  x  <->  A  e.  y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289   E.wex 1426    e. wcel 1438
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472
This theorem depends on definitions:  df-bi 115  df-clel 2084
This theorem is referenced by: (None)
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