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Theorem cbvrex2v 2946
Description: Change bound variables of double restricted universal quantification, using implicit substitution. (Contributed by FL, 2-Jul-2012.)
Hypotheses
Ref Expression
cbvrex2v.1  |-  ( x  =  z  ->  ( ph 
<->  ch ) )
cbvrex2v.2  |-  ( y  =  w  ->  ( ch 
<->  ps ) )
Assertion
Ref Expression
cbvrex2v  |-  ( E. x  e.  A  E. y  e.  B  ph  <->  E. z  e.  A  E. w  e.  B  ps )
Distinct variable groups:    x, A    z, A    w, B    x, B, y    z, B, y    ch, w    ch, x    ph, z    ps, y
Allowed substitution hints:    ph( x, y, w)    ps( x, z, w)    ch( y, z)    A( y, w)

Proof of Theorem cbvrex2v
StepHypRef Expression
1 cbvrex2v.1 . . . 4  |-  ( x  =  z  ->  ( ph 
<->  ch ) )
21rexbidv 2726 . . 3  |-  ( x  =  z  ->  ( E. y  e.  B  ph  <->  E. y  e.  B  ch ) )
32cbvrexv 2938 . 2  |-  ( E. x  e.  A  E. y  e.  B  ph  <->  E. z  e.  A  E. y  e.  B  ch )
4 cbvrex2v.2 . . . 4  |-  ( y  =  w  ->  ( ch 
<->  ps ) )
54cbvrexv 2938 . . 3  |-  ( E. y  e.  B  ch  <->  E. w  e.  B  ps )
65rexbii 2730 . 2  |-  ( E. z  e.  A  E. y  e.  B  ch  <->  E. z  e.  A  E. w  e.  B  ps )
73, 6bitri 249 1  |-  ( E. x  e.  A  E. y  e.  B  ph  <->  E. z  e.  A  E. w  e.  B  ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   E.wrex 2706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ral 2710  df-rex 2711
This theorem is referenced by:  omeu  7012  oeeui  7029  eroveu  7183  genpv  9155  bezoutlem3  13706  bezoutlem4  13707  bezout  13708  4sqlem2  13992  vdwnn  14041  efgrelexlema  16225  dyadmax  20919  2sqlem9  22596  2sq  22599  legov  22891  pstmfval  26176  nn0prpwlem  28358  isbnd2  28523
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