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Theorem rexrot4 2816
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 2815 . . 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 2676 . 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 2815 . 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 241 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 177   E.wrex 2652
This theorem is referenced by:  lsmspsn  16085
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-cleq 2382  df-clel 2385  df-nfc 2514  df-rex 2657
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