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Theorem cbvrex2v 2692
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 2458 . . 3  |-  ( x  =  z  ->  ( E. y  e.  B  ph  <->  E. y  e.  B  ch ) )
32cbvrexv 2681 . 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 2681 . . 3  |-  ( E. y  e.  B  ch  <->  E. w  e.  B  ps )
65rexbii 2464 . 2  |-  ( E. z  e.  A  E. y  e.  B  ch  <->  E. z  e.  A  E. w  e.  B  ps )
73, 6bitri 183 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 104   E.wrex 2436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rex 2441
This theorem is referenced by:  eroveu  6572  genipv  7430  bezoutlemnewy  11884  xmettx  12952
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