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Theorem rexcom13 3029
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 3028 . 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 3028 . . 3  |-  ( E. x  e.  A  E. z  e.  C  ph  <->  E. z  e.  C  E. x  e.  A  ph )
32rexbii 2969 . 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 3028 . 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 2818
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1597  df-nf 1600  df-sb 1712  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2822  df-rex 2823
This theorem is referenced by:  rexrot4  3030
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