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Theorem rexcom13 2878
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 2877 . 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 2877 . . 3  |-  ( E. x  e.  A  E. z  e.  C  ph  <->  E. z  e.  C  E. x  e.  A  ph )
32rexbii 2735 . 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 2877 . 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 271 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 setvar class
Syntax hints:    <-> wb 184   E.wrex 2711
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rex 2716
This theorem is referenced by:  rexrot4  2879
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