MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rexrot4 Unicode version

Theorem rexrot4 2703
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 2702 . . 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 2568 . 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 2702 . 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 2544
This theorem is referenced by:  lsmspsn  15837
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549
  Copyright terms: Public domain W3C validator