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Theorem rexcom13 2715
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 2714 . 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 2714 . . 3  |-  ( E. x  e.  A  E. z  e.  C  ph  <->  E. z  e.  C  E. x  e.  A  ph )
32rexbii 2581 . 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 2714 . 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 2557
This theorem is referenced by:  rexrot4  2716
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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