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Theorem rexcom13 2702
Description: Swap 1st and 3rd restricted existential quantifiers. (Contributed by NM, 8-Apr-2015.)
Assertion
Ref Expression
rexcom13  |-  ( E. x  e.  A  E. y  e.  B  E. z  e.  C  ph  <->  E. z  e.  C  E. y  e.  B  E. x  e.  A  ph )
Distinct variable groups:    y, z, A    x, z, B    x, y, C
Allowed substitution hints:    ph( x, y, z)    A( x)    B( y)    C( z)

Proof of Theorem rexcom13
StepHypRef Expression
1 rexcom 2701 . 2  |-  ( E. x  e.  A  E. y  e.  B  E. z  e.  C  ph  <->  E. y  e.  B  E. x  e.  A  E. z  e.  C  ph )
2 rexcom 2701 . . 3  |-  ( E. x  e.  A  E. z  e.  C  ph  <->  E. z  e.  C  E. x  e.  A  ph )
32rexbii 2568 . 2  |-  ( E. y  e.  B  E. x  e.  A  E. z  e.  C  ph  <->  E. y  e.  B  E. z  e.  C  E. x  e.  A  ph )
4 rexcom 2701 . 2  |-  ( E. y  e.  B  E. z  e.  C  E. x  e.  A  ph  <->  E. z  e.  C  E. y  e.  B  E. x  e.  A  ph )
51, 3, 43bitri 262 1  |-  ( E. x  e.  A  E. y  e.  B  E. z  e.  C  ph  <->  E. z  e.  C  E. y  e.  B  E. x  e.  A  ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   E.wrex 2544
This theorem is referenced by:  rexrot4  2703
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
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