Theorem reuhyp 2900

Description: A theorem useful for eliminating the restricted existential uniqueness hypotheses in reuxfr 2899.

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 groups:   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 1813	. . . . 5 |- (B e. C -> B e. V)
3	1, 2	syl 10	. . . 4 |- (x e. C -> B e. V)
4		eueq 1912	. . . 4 |- (B e. V <-> E!y y = B)
5	3, 4	sylib 198	. . 3 |- (x e. C -> E!y y = B)
6		eleq1 1531	. . . . . . 7 |- (y = B -> (y e. C <-> B e. C))
7	6, 1	syl5cbir 211	. . . . . 6 |- (x e. C -> (y = B -> y e. C))
8	7	pm4.71rd 638	. . . . 5 |- (x e. C -> (y = B <-> (y e. C /\ y = B)))
9		reuhyp.2	. . . . . 6 |- ((x e. C /\ y e. C) -> (x = A <-> y = B))
10	9	pm5.32da 648	. . . . 5 |- (x e. C -> ((y e. C /\ x = A) <-> (y e. C /\ y = B)))
11	8, 10	bitr4d 530	. . . 4 |- (x e. C -> (y = B <-> (y e. C /\ x = A)))
12	11	eubidv 1384	. . 3 |- (x e. C -> (E!y y = B <-> E!y(y e. C /\ x = A)))
13	5, 12	mpbid 195	. 2 |- (x e. C -> E!y(y e. C /\ x = A))
14		df-reu 1648	. 2 |- (E!y e. C x = A <-> E!y(y e. C /\ x = A))
15	13, 14	sylibr 200	1 |- (x e. C -> E!y e. C x = A)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 954   e. wcel 956  E!weu 1378  E!wreu 1644  Vcvv 1807

This theorem is referenced by:  reuunineg 6021  zmax 6176  rebtwnz 6178

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-reu 1648  df-v 1808

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