Theorem rexcom13 2989

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 2988	. 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 2988	. . 3 |- ( E. x e. A E. z e. C ph <-> E. z e. C E. x e. A ph )
3	2	rexbii 2862	. 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 2988	. 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 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 )

Syntax hints:   <-> wb 184   E.wrex 2800

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rex 2805

This theorem is referenced by:  rexrot4  2990