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Theorem cbvrexvw 2702
Description: Version of cbvrexv 2698 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.) Reduce axiom usage. (Revised by Gino Giotto, 25-Aug-2024.)
Hypothesis
Ref Expression
cbvralvw.1  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
cbvrexvw  |-  ( E. x  e.  A  ph  <->  E. y  e.  A  ps )
Distinct variable groups:    x, y, A    ph, y    ps, x
Allowed substitution hints:    ph( x)    ps( y)

Proof of Theorem cbvrexvw
StepHypRef Expression
1 eleq1w 2232 . . . 4  |-  ( x  =  y  ->  (
x  e.  A  <->  y  e.  A ) )
2 cbvralvw.1 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
31, 2anbi12d 471 . . 3  |-  ( x  =  y  ->  (
( x  e.  A  /\  ph )  <->  ( y  e.  A  /\  ps )
) )
43cbvexvw 1914 . 2  |-  ( E. x ( x  e.  A  /\  ph )  <->  E. y ( y  e.  A  /\  ps )
)
5 df-rex 2455 . 2  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
6 df-rex 2455 . 2  |-  ( E. y  e.  A  ps  <->  E. y ( y  e.  A  /\  ps )
)
74, 5, 63bitr4i 211 1  |-  ( E. x  e.  A  ph  <->  E. y  e.  A  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   E.wex 1486    e. wcel 2142   E.wrex 2450
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-5 1441  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528
This theorem depends on definitions:  df-bi 116  df-tru 1352  df-nf 1455  df-clel 2167  df-rex 2455
This theorem is referenced by:  cbvrex2vw  2709  prodmodclem2  11544  prodmodc  11545  zsupssdc  11913  pceu  12253  grpridd  12645  dfgrp2  12736  dfgrp3mlem  12801  bj-charfunbi  13931
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