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Theorem rexrot4 2600
Description: Rotate existential restricted quantifiers twice. (Contributed by NM, 8-Apr-2015.)
Assertion
Ref Expression
rexrot4  |-  ( E. x  e.  A  E. y  e.  B  E. z  e.  C  E. w  e.  D  ph  <->  E. z  e.  C  E. w  e.  D  E. x  e.  A  E. y  e.  B  ph )
Distinct variable groups:    z, w, A   
w, B, z    x, w, y, C    x, z, D, y
Allowed substitution hints:    ph( x, y, z, w)    A( x, y)    B( x, y)    C( z)    D( w)

Proof of Theorem rexrot4
StepHypRef Expression
1 rexcom13 2599 . . 3  |-  ( E. y  e.  B  E. z  e.  C  E. w  e.  D  ph  <->  E. w  e.  D  E. z  e.  C  E. y  e.  B  ph )
21rexbii 2445 . 2  |-  ( E. x  e.  A  E. y  e.  B  E. z  e.  C  E. w  e.  D  ph  <->  E. x  e.  A  E. w  e.  D  E. z  e.  C  E. y  e.  B  ph )
3 rexcom13 2599 . 2  |-  ( E. x  e.  A  E. w  e.  D  E. z  e.  C  E. y  e.  B  ph  <->  E. z  e.  C  E. w  e.  D  E. x  e.  A  E. y  e.  B  ph )
42, 3bitri 183 1  |-  ( E. x  e.  A  E. y  e.  B  E. z  e.  C  E. w  e.  D  ph  <->  E. z  e.  C  E. w  e.  D  E. x  e.  A  E. y  e.  B  ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 104   E.wrex 2418
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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423
This theorem is referenced by: (None)
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