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Theorem cbvrex2v 2773
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 2564 . . 3  |-  ( x  =  z  ->  ( E. y  e.  B  ph  <->  E. y  e.  B  ch ) )
32cbvrexv 2765 . 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 2765 . . 3  |-  ( E. y  e.  B  ch  <->  E. w  e.  B  ps )
65rexbii 2568 . 2  |-  ( E. z  e.  A  E. y  e.  B  ch  <->  E. z  e.  A  E. w  e.  B  ps )
73, 6bitri 240 1  |-  ( E. x  e.  A  E. y  e.  B  ph  <->  E. z  e.  A  E. w  e.  B  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   E.wrex 2544
This theorem is referenced by:  omeu  6583  oeeui  6600  eroveu  6753  genpv  8623  bezoutlem3  12719  bezoutlem4  12720  bezout  12721  4sqlem2  12996  vdwnn  13045  efgrelexlema  15058  dyadmax  18953  2sqlem9  20612  2sq  20615  nn0prpwlem  26238  isbnd2  26507
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-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549
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