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

Step	Hyp	Ref	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 )
3	2	rexbii 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 )
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 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