Theorem rexcom13 2519

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

Step	Hyp	Ref	Expression
1		rexcom 2518	. 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 2518	. . 3 |- ( E. x e. A E. z e. C ph <-> E. z e. C E. x e. A ph )
3	2	rexbii 2373	. 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 2518	. 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 )
5	1, 3, 4	3bitri 204	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 )

Syntax hints:   <-> wb 103   E.wrex 2349

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354

This theorem is referenced by:  rexrot4  2520