Theorem 3reeanv 2779

Description: Rearrange three existential quantifiers. (Contributed by Jeff Madsen, 11-Jun-2010.)

Assertion

Ref	Expression
3reeanv	|- (E.x e. A E.y e. B E.z e. C (ph /\ ps /\ ch) <-> (E.x e. A ph /\ E.y e. B ps /\ E.z e. C ch))

Distinct variable groups:   ph,y,z   ps,x,z   ch,x,y   y,A   x,B,z   x,C,y

Allowed substitution hints:   ph(x)   ps(y)   ch(z)   A(x,z)   B(y)   C(z)

Proof of Theorem 3reeanv

Step	Hyp	Ref	Expression
1		r19.41v 2764	. . 3 |- (E.x e. A (E.y e. B (ph /\ ps) /\ E.z e. C ch) <-> (E.x e. A E.y e. B (ph /\ ps) /\ E.z e. C ch))
2		reeanv 2778	. . . 4 |- (E.x e. A E.y e. B (ph /\ ps) <-> (E.x e. A ph /\ E.y e. B ps))
3	2	anbi1i 676	. . 3 |- ((E.x e. A E.y e. B (ph /\ ps) /\ E.z e. C ch) <-> ((E.x e. A ph /\ E.y e. B ps) /\ E.z e. C ch))
4	1, 3	bitri 240	. 2 |- (E.x e. A (E.y e. B (ph /\ ps) /\ E.z e. C ch) <-> ((E.x e. A ph /\ E.y e. B ps) /\ E.z e. C ch))
5		df-3an 936	. . . . 5 |- ((ph /\ ps /\ ch) <-> ((ph /\ ps) /\ ch))
6	5	2rexbii 2641	. . . 4 |- (E.y e. B E.z e. C (ph /\ ps /\ ch) <-> E.y e. B E.z e. C ((ph /\ ps) /\ ch))
7		reeanv 2778	. . . 4 |- (E.y e. B E.z e. C ((ph /\ ps) /\ ch) <-> (E.y e. B (ph /\ ps) /\ E.z e. C ch))
8	6, 7	bitri 240	. . 3 |- (E.y e. B E.z e. C (ph /\ ps /\ ch) <-> (E.y e. B (ph /\ ps) /\ E.z e. C ch))
9	8	rexbii 2639	. 2 |- (E.x e. A E.y e. B E.z e. C (ph /\ ps /\ ch) <-> E.x e. A (E.y e. B (ph /\ ps) /\ E.z e. C ch))
10		df-3an 936	. 2 |- ((E.x e. A ph /\ E.y e. B ps /\ E.z e. C ch) <-> ((E.x e. A ph /\ E.y e. B ps) /\ E.z e. C ch))
11	4, 9, 10	3bitr4i 268	1 |- (E.x e. A E.y e. B E.z e. C (ph /\ ps /\ ch) <-> (E.x e. A ph /\ E.y e. B ps /\ E.z e. C ch))

Syntax hints:   <-> wb 176   /\ wa 358   /\ w3a 934  E.wrex 2615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rex 2620

This theorem is referenced by:  xpassen  6057