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Theorem cbvraldva2 2928
Description: Rule used to change the bound variable in a restricted universal 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
cbvraldva2  |-  ( ph  ->  ( A. x  e.  A  ps  <->  A. 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 cbvraldva2
StepHypRef Expression
1 simpr 448 . . . . 5  |-  ( (
ph  /\  x  =  y )  ->  x  =  y )
2 cbvraldva2.2 . . . . 5  |-  ( (
ph  /\  x  =  y )  ->  A  =  B )
31, 2eleq12d 2503 . . . 4  |-  ( (
ph  /\  x  =  y )  ->  (
x  e.  A  <->  y  e.  B ) )
4 cbvraldva2.1 . . . 4  |-  ( (
ph  /\  x  =  y )  ->  ( ps 
<->  ch ) )
53, 4imbi12d 312 . . 3  |-  ( (
ph  /\  x  =  y )  ->  (
( x  e.  A  ->  ps )  <->  ( y  e.  B  ->  ch )
) )
65cbvaldva 1994 . 2  |-  ( ph  ->  ( A. x ( x  e.  A  ->  ps )  <->  A. y ( y  e.  B  ->  ch ) ) )
7 df-ral 2702 . 2  |-  ( A. x  e.  A  ps  <->  A. x ( x  e.  A  ->  ps )
)
8 df-ral 2702 . 2  |-  ( A. y  e.  B  ch  <->  A. y ( y  e.  B  ->  ch )
)
96, 7, 83bitr4g 280 1  |-  ( ph  ->  ( A. x  e.  A  ps  <->  A. y  e.  B  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1549    = wceq 1652    e. wcel 1725   A.wral 2697
This theorem is referenced by:  cbvraldva  2930  mreexexlemd  13861
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-nf 1554  df-cleq 2428  df-clel 2431  df-ral 2702
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