Theorem reu3 3026

Description: A way to express restricted uniqueness. (Contributed by NM, 24-Oct-2006.)

Assertion

Ref	Expression
reu3	|- (E!x e. A ph <-> (E.x e. A ph /\ E.y e. A A.x e. A (ph -> x = y)))

Distinct variable groups:   x,y,A   ph,y

Allowed substitution hint:   ph(x)

Proof of Theorem reu3

Step	Hyp	Ref	Expression
1		reurex 2825	. . 3 |- (E!x e. A ph -> E.x e. A ph)
2		reu6 3025	. . . 4 |- (E!x e. A ph <-> E.y e. A A.x e. A (ph <-> x = y))
3		bi1 178	. . . . . 6 |- ((ph <-> x = y) -> (ph -> x = y))
4	3	ralimi 2689	. . . . 5 |- (A.x e. A (ph <-> x = y) -> A.x e. A (ph -> x = y))
5	4	reximi 2721	. . . 4 |- (E.y e. A A.x e. A (ph <-> x = y) -> E.y e. A A.x e. A (ph -> x = y))
6	2, 5	sylbi 187	. . 3 |- (E!x e. A ph -> E.y e. A A.x e. A (ph -> x = y))
7	1, 6	jca 518	. 2 |- (E!x e. A ph -> (E.x e. A ph /\ E.y e. A A.x e. A (ph -> x = y)))
8		rexex 2673	. . . 4 |- (E.y e. A A.x e. A (ph -> x = y) -> E.yA.x e. A (ph -> x = y))
9	8	anim2i 552	. . 3 |- ((E.x e. A ph /\ E.y e. A A.x e. A (ph -> x = y)) -> (E.x e. A ph /\ E.yA.x e. A (ph -> x = y)))
10		nfv 1619	. . . . 5 |- F/y(x e. A /\ ph)
11	10	eu3 2230	. . . 4 |- (E!x(x e. A /\ ph) <-> (E.x(x e. A /\ ph) /\ E.yA.x((x e. A /\ ph) -> x = y)))
12		df-reu 2621	. . . 4 |- (E!x e. A ph <-> E!x(x e. A /\ ph))
13		df-rex 2620	. . . . 5 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
14		df-ral 2619	. . . . . . 7 |- (A.x e. A (ph -> x = y) <-> A.x(x e. A -> (ph -> x = y)))
15		impexp 433	. . . . . . . 8 |- (((x e. A /\ ph) -> x = y) <-> (x e. A -> (ph -> x = y)))
16	15	albii 1566	. . . . . . 7 |- (A.x((x e. A /\ ph) -> x = y) <-> A.x(x e. A -> (ph -> x = y)))
17	14, 16	bitr4i 243	. . . . . 6 |- (A.x e. A (ph -> x = y) <-> A.x((x e. A /\ ph) -> x = y))
18	17	exbii 1582	. . . . 5 |- (E.yA.x e. A (ph -> x = y) <-> E.yA.x((x e. A /\ ph) -> x = y))
19	13, 18	anbi12i 678	. . . 4 |- ((E.x e. A ph /\ E.yA.x e. A (ph -> x = y)) <-> (E.x(x e. A /\ ph) /\ E.yA.x((x e. A /\ ph) -> x = y)))
20	11, 12, 19	3bitr4i 268	. . 3 |- (E!x e. A ph <-> (E.x e. A ph /\ E.yA.x e. A (ph -> x = y)))
21	9, 20	sylibr 203	. 2 |- ((E.x e. A ph /\ E.y e. A A.x e. A (ph -> x = y)) -> E!x e. A ph)
22	7, 21	impbii 180	1 |- (E!x e. A ph <-> (E.x e. A ph /\ E.y e. A A.x e. A (ph -> x = y)))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540  E.wex 1541   = wceq 1642   e. wcel 1710  E!weu 2204  A.wral 2614  E.wrex 2615  E!wreu 2616

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-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-cleq 2346  df-clel 2349  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622

This theorem is referenced by:  reu7  3031