Theorem rexcom13 2635

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 2634	. 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 2634	. . 3 |- ( E. x e. A E. z e. C ph <-> E. z e. C E. x e. A ph )
3	2	rexbii 2477	. 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 2634	. 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 205	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 104   E.wrex 2449

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454

This theorem is referenced by:  rexrot4  2636