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Theorem rexrot4 2990
Description: Rotate four restricted existential 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 2989 . . 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 2925 . 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 2989 . 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 252 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 setvar class
Syntax hints:    <-> wb 187   E.wrex 2774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-ex 1660  df-nf 1664  df-sb 1787  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ral 2778  df-rex 2779
This theorem is referenced by:  lsmspsn  18235
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