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Theorem rexxfr2d 4514
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 294 . . . 4  |-  ( (
ph  /\  x  =  A )  ->  ( -.  ps  <->  -.  ch )
)
51, 2, 4ralxfr2d 4513 . . 3  |-  ( ph  ->  ( A. x  e.  B  -.  ps  <->  A. y  e.  C  -.  ch )
)
65notbid 294 . 2  |-  ( ph  ->  ( -.  A. x  e.  B  -.  ps  <->  -.  A. y  e.  C  -.  ch )
)
7 dfrex2 2733 . 2  |-  ( E. x  e.  B  ps  <->  -. 
A. x  e.  B  -.  ps )
8 dfrex2 2733 . 2  |-  ( E. y  e.  C  ch  <->  -. 
A. y  e.  C  -.  ch )
96, 7, 83bitr4g 288 1  |-  ( ph  ->  ( E. x  e.  B  ps  <->  E. y  e.  C  ch )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2720   E.wrex 2721
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ral 2725  df-rex 2726  df-v 2979
This theorem is referenced by:  rexrn  5850  rexima  5961  cnpresti  18897  cnprest  18898  1stcrest  19062  subislly  19090  txrest  19209  trfil2  19465  met1stc  20101  xrlimcnp  22367  esumlub  26516  esumfsup  26524  ptrest  28430  djhcvat42  35065
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