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Theorem cbvreuvw 2702
Description: Version of cbvreuv 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
cbvreuvw  |-  ( 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 cbvreuvw
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 eleq1w 2231 . . . . . . 7  |-  ( x  =  y  ->  (
x  e.  A  <->  y  e.  A ) )
2 cbvralvw.1 . . . . . . 7  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
31, 2anbi12d 470 . . . . . 6  |-  ( x  =  y  ->  (
( x  e.  A  /\  ph )  <->  ( y  e.  A  /\  ps )
) )
4 equequ1 1705 . . . . . 6  |-  ( x  =  y  ->  (
x  =  z  <->  y  =  z ) )
53, 4bibi12d 234 . . . . 5  |-  ( x  =  y  ->  (
( ( x  e.  A  /\  ph )  <->  x  =  z )  <->  ( (
y  e.  A  /\  ps )  <->  y  =  z ) ) )
65cbvalvw 1912 . . . 4  |-  ( A. x ( ( x  e.  A  /\  ph ) 
<->  x  =  z )  <->  A. y ( ( y  e.  A  /\  ps ) 
<->  y  =  z ) )
76exbii 1598 . . 3  |-  ( E. z A. x ( ( x  e.  A  /\  ph )  <->  x  =  z )  <->  E. z A. y ( ( y  e.  A  /\  ps ) 
<->  y  =  z ) )
8 df-eu 2022 . . 3  |-  ( E! x ( x  e.  A  /\  ph )  <->  E. z A. x ( ( x  e.  A  /\  ph )  <->  x  =  z ) )
9 df-eu 2022 . . 3  |-  ( E! y ( y  e.  A  /\  ps )  <->  E. z A. y ( ( y  e.  A  /\  ps )  <->  y  =  z ) )
107, 8, 93bitr4ri 212 . 2  |-  ( E! y ( y  e.  A  /\  ps )  <->  E! x ( x  e.  A  /\  ph )
)
11 df-reu 2455 . 2  |-  ( E! y  e.  A  ps  <->  E! y ( y  e.  A  /\  ps )
)
12 df-reu 2455 . 2  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
1310, 11, 123bitr4ri 212 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   A.wal 1346   E.wex 1485   E!weu 2019    e. wcel 2141   E!wreu 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 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-eu 2022  df-clel 2166  df-reu 2455
This theorem is referenced by: (None)
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