Theorem reu6 1932

Description: A way to express restricted uniqueness.

Assertion

Ref	Expression
reu6	|- (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

Proof of Theorem reu6

Step	Hyp	Ref	Expression
1		reurex 1928	. . 3 |- (E!x e. A ph -> E.x e. A ph)
2		reu3 1931	. . . 4 |- (E!x e. A ph <-> E.y e. A A.x e. A (ph <-> x = y))
3		bi1 148	. . . . . 6 |- ((ph <-> x = y) -> (ph -> x = y))
4	3	r19.20si 1706	. . . . 5 |- (A.x e. A (ph <-> x = y) -> A.x e. A (ph -> x = y))
5	4	r19.22si 1734	. . . 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 199	. . 3 |- (E!x e. A ph -> E.y e. A A.x e. A (ph -> x = y))
7	1, 6	jca 288	. 2 |- (E!x e. A ph -> (E.x e. A ph /\ E.y e. A A.x e. A (ph -> x = y)))
8		rexex 1693	. . . 4 |- (E.y e. A A.x e. A (ph -> x = y) -> E.yA.x e. A (ph -> x = y))
9	8	anim2i 335	. . 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		ax-17 971	. . . . 5 |- ((x e. A /\ ph) -> A.y(x e. A /\ ph))
11	10	eu3 1397	. . . 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 1651	. . . 4 |- (E!x e. A ph <-> E!x(x e. A /\ ph))
13		df-rex 1650	. . . . 5 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
14		df-ral 1649	. . . . . . 7 |- (A.x e. A (ph -> x = y) <-> A.x(x e. A -> (ph -> x = y)))
15		impexp 347	. . . . . . . 8 |- (((x e. A /\ ph) -> x = y) <-> (x e. A -> (ph -> x = y)))
16	15	albii 999	. . . . . . 7 |- (A.x((x e. A /\ ph) -> x = y) <-> A.x(x e. A -> (ph -> x = y)))
17	14, 16	bitr4 176	. . . . . 6 |- (A.x e. A (ph -> x = y) <-> A.x((x e. A /\ ph) -> x = y))
18	17	exbii 1051	. . . . 5 |- (E.yA.x e. A (ph -> x = y) <-> E.yA.x((x e. A /\ ph) -> x = y))
19	13, 18	anbi12i 482	. . . 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	3bitr4 183	. . 3 |- (E!x e. A ph <-> (E.x e. A ph /\ E.yA.x e. A (ph -> x = y)))
21	9, 20	sylibr 200	. 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	impbi 157	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 3   <-> wb 146   /\ wa 223  A.wal 954   = wceq 956   e. wcel 958  E.wex 980  E!weu 1380  A.wral 1645  E.wrex 1646  E!wreu 1647

This theorem is referenced by:  reu7 1935

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-cleq 1469  df-clel 1472  df-ral 1649  df-rex 1650  df-reu 1651

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