Theorem reuhyp 1581

Description: A theorem useful for eliminating the restricted existential uniqueness hypotheses in reuxfr 1580.

Hypotheses

Ref	Expression
reuhyp.1	|- (x e. C -> B e. C)
reuhyp.2	|- ((x e. C /\ y e. C) -> (x = A <-> y = B))

Assertion

Ref	Expression
reuhyp	|- (x e. C -> E!y e. C x = A)

Distinct variable group(s):   x,A   y,B   y,C   x,y

Proof of Theorem reuhyp

Step	Hyp	Ref	Expression
1		reuhyp.1	. . . . 5 |- (x e. C -> B e. C)
2		elisset 1354	. . . . 5 |- (B e. C -> B e. V)
3	1, 2	syl 12	. . . 4 |- (x e. C -> B e. V)
4		eueq 1427	. . . 4 |- (B e. V <-> E!y y = B)
5	3, 4	sylib 173	. . 3 |- (x e. C -> E!y y = B)
6		eleq1 1149	. . . . . . . 8 |- (y = B -> (y e. C <-> B e. C))
7	6, 1	syl5bir 184	. . . . . . 7 |- (y = B -> (x e. C -> y e. C))
8	7	com12 13	. . . . . 6 |- (x e. C -> (y = B -> y e. C))
9		pm4.71r 482	. . . . . 6 |- ((y = B -> y e. C) <-> (y = B <-> (y e. C /\ y = B)))
10	8, 9	sylib 173	. . . . 5 |- (x e. C -> (y = B <-> (y e. C /\ y = B)))
11		reuhyp.2	. . . . . . 7 |- ((x e. C /\ y e. C) -> (x = A <-> y = B))
12	11	exp 291	. . . . . 6 |- (x e. C -> (y e. C -> (x = A <-> y = B)))
13	12	pm5.32d 491	. . . . 5 |- (x e. C -> ((y e. C /\ x = A) <-> (y e. C /\ y = B)))
14	10, 13	bitr4d 409	. . . 4 |- (x e. C -> (y = B <-> (y e. C /\ x = A)))
15	14	bieudv 1013	. . 3 |- (x e. C -> (E!y y = B <-> E!y(y e. C /\ x = A)))
16	5, 15	mpbid 170	. 2 |- (x e. C -> E!y(y e. C /\ x = A))
17		df-reu 1207	. 2 |- (E!y e. C x = A <-> E!y(y e. C /\ x = A))
18	16, 17	sylibr 175	1 |- (x e. C -> E!y e. C x = A)

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  E!weu 1007   = wceq 1091   e. wcel 1092  E!wreu 1203  Vcvv 1348

This theorem is referenced by:  zmax 4618  rebtwnz 4620

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-reu 1207  df-v 1349

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