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Theorem rexrot4 2716
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 2715 . . 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 2581 . 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 2715 . 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 240 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 176   E.wrex 2557
This theorem is referenced by:  lsmspsn  15853
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562
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