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Theorem cbvrex2v 3018
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 2893 . . 3  |-  ( x  =  z  ->  ( E. y  e.  B  ph  <->  E. y  e.  B  ch ) )
32cbvrexv 3010 . 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 3010 . . 3  |-  ( E. y  e.  B  ch  <->  E. w  e.  B  ps )
65rexbii 2884 . 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 2733
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-ex 1621  df-nf 1625  df-sb 1748  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ral 2737  df-rex 2738
This theorem is referenced by:  omeu  7152  oeeui  7169  eroveu  7324  genpv  9288  bezoutlem3  14180  bezoutlem4  14181  bezout  14182  4sqlem2  14469  vdwnn  14518  efgrelexlema  16884  dyadmax  22092  2sqlem9  23765  2sq  23768  legov  24092  pstmfval  28029  nn0prpwlem  30306  isbnd2  30445  fourierdlem42  32097  fourierdlem54  32109
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