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Theorem rexxfr2d 4680
Description: Transfer universal quantification from a variable  x to another variable  y contained in expression  A. (Contributed by Mario Carneiro, 20-Aug-2014.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
Hypotheses
Ref Expression
ralxfr2d.1  |-  ( (
ph  /\  y  e.  C )  ->  A  e.  V )
ralxfr2d.2  |-  ( ph  ->  ( x  e.  B  <->  E. y  e.  C  x  =  A ) )
ralxfr2d.3  |-  ( (
ph  /\  x  =  A )  ->  ( ps 
<->  ch ) )
Assertion
Ref Expression
rexxfr2d  |-  ( ph  ->  ( E. x  e.  B  ps  <->  E. y  e.  C  ch )
)
Distinct variable groups:    x, A    x, y, B    x, C    ch, x    ph, x, y    ps, y
Allowed substitution hints:    ps( x)    ch( y)    A( y)    C( y)    V( x, y)

Proof of Theorem rexxfr2d
StepHypRef Expression
1 ralxfr2d.1 . . . 4  |-  ( (
ph  /\  y  e.  C )  ->  A  e.  V )
2 ralxfr2d.2 . . . 4  |-  ( ph  ->  ( x  e.  B  <->  E. y  e.  C  x  =  A ) )
3 ralxfr2d.3 . . . . 5  |-  ( (
ph  /\  x  =  A )  ->  ( ps 
<->  ch ) )
43notbid 286 . . . 4  |-  ( (
ph  /\  x  =  A )  ->  ( -.  ps  <->  -.  ch )
)
51, 2, 4ralxfr2d 4679 . . 3  |-  ( ph  ->  ( A. x  e.  B  -.  ps  <->  A. y  e.  C  -.  ch )
)
65notbid 286 . 2  |-  ( ph  ->  ( -.  A. x  e.  B  -.  ps  <->  -.  A. y  e.  C  -.  ch )
)
7 dfrex2 2662 . 2  |-  ( E. x  e.  B  ps  <->  -. 
A. x  e.  B  -.  ps )
8 dfrex2 2662 . 2  |-  ( E. y  e.  C  ch  <->  -. 
A. y  e.  C  -.  ch )
96, 7, 83bitr4g 280 1  |-  ( ph  ->  ( E. x  e.  B  ps  <->  E. y  e.  C  ch )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2649   E.wrex 2650
This theorem is referenced by:  rexrn  5811  rexima  5916  cnpresti  17274  cnprest  17275  1stcrest  17437  subislly  17465  txrest  17584  trfil2  17840  met1stc  18441  xrlimcnp  20674  esumlub  24248  esumfsup  24256  djhcvat42  31530
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ral 2654  df-rex 2655  df-v 2901
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