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Theorem cbvrexdva2 2769
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 447 . . . . 5  |-  ( (
ph  /\  x  =  y )  ->  x  =  y )
2 cbvraldva2.2 . . . . 5  |-  ( (
ph  /\  x  =  y )  ->  A  =  B )
31, 2eleq12d 2351 . . . 4  |-  ( (
ph  /\  x  =  y )  ->  (
x  e.  A  <->  y  e.  B ) )
4 cbvraldva2.1 . . . 4  |-  ( (
ph  /\  x  =  y )  ->  ( ps 
<->  ch ) )
53, 4anbi12d 691 . . 3  |-  ( (
ph  /\  x  =  y )  ->  (
( x  e.  A  /\  ps )  <->  ( y  e.  B  /\  ch )
) )
65cbvexdva 1951 . 2  |-  ( ph  ->  ( E. x ( x  e.  A  /\  ps )  <->  E. y ( y  e.  B  /\  ch ) ) )
7 df-rex 2549 . 2  |-  ( E. x  e.  A  ps  <->  E. x ( x  e.  A  /\  ps )
)
8 df-rex 2549 . 2  |-  ( E. y  e.  B  ch  <->  E. y ( y  e.  B  /\  ch )
)
96, 7, 83bitr4g 279 1  |-  ( ph  ->  ( E. x  e.  A  ps  <->  E. y  e.  B  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   E.wrex 2544
This theorem is referenced by:  cbvrexdva  2771  mreexexlemd  13546
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529  df-nf 1532  df-cleq 2276  df-clel 2279  df-rex 2549
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