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Theorem cbvrexdva2 2636
Description: Rule used to change the bound variable in a restricted existential quantifier with implicit substitution which also changes the quantifier domain. Deduction form. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
cbvraldva2.1  |-  ( (
ph  /\  x  =  y )  ->  ( ps 
<->  ch ) )
cbvraldva2.2  |-  ( (
ph  /\  x  =  y )  ->  A  =  B )
Assertion
Ref Expression
cbvrexdva2  |-  ( ph  ->  ( E. x  e.  A  ps  <->  E. y  e.  B  ch )
)
Distinct variable groups:    y, A    ps, y    x, B    ch, x    ph, x, y
Allowed substitution hints:    ps( x)    ch( y)    A( x)    B( y)

Proof of Theorem cbvrexdva2
StepHypRef Expression
1 simpr 109 . . . . 5  |-  ( (
ph  /\  x  =  y )  ->  x  =  y )
2 cbvraldva2.2 . . . . 5  |-  ( (
ph  /\  x  =  y )  ->  A  =  B )
31, 2eleq12d 2188 . . . 4  |-  ( (
ph  /\  x  =  y )  ->  (
x  e.  A  <->  y  e.  B ) )
4 cbvraldva2.1 . . . 4  |-  ( (
ph  /\  x  =  y )  ->  ( ps 
<->  ch ) )
53, 4anbi12d 464 . . 3  |-  ( (
ph  /\  x  =  y )  ->  (
( x  e.  A  /\  ps )  <->  ( y  e.  B  /\  ch )
) )
65cbvexdva 1881 . 2  |-  ( ph  ->  ( E. x ( x  e.  A  /\  ps )  <->  E. y ( y  e.  B  /\  ch ) ) )
7 df-rex 2399 . 2  |-  ( E. x  e.  A  ps  <->  E. x ( x  e.  A  /\  ps )
)
8 df-rex 2399 . 2  |-  ( E. y  e.  B  ch  <->  E. y ( y  e.  B  /\  ch )
)
96, 7, 83bitr4g 222 1  |-  ( ph  ->  ( E. x  e.  A  ps  <->  E. y  e.  B  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1316   E.wex 1453    e. wcel 1465   E.wrex 2394
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 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-cleq 2110  df-clel 2113  df-rex 2399
This theorem is referenced by:  cbvrexdva  2638  acexmid  5741
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