Theorem reuhypd 3848

Description: A theorem useful for eliminating the restricted existential uniqueness hypotheses in riotaxfrd 5581.

Hypotheses

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

Assertion

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

Distinct variable groups:   ph,y   y,B   y,C   x,y

Proof of Theorem reuhypd

Step	Hyp	Ref	Expression
1		reuhypd.1	. . . . 5 |- ((ph /\ x e. C) -> B e. C)
2		elisset 2299	. . . . 5 |- (B e. C -> B e. _V)
3	1, 2	syl 12	. . . 4 |- ((ph /\ x e. C) -> B e. _V)
4		eueq 2427	. . . 4 |- (B e. _V <-> E!y y = B)
5	3, 4	sylib 215	. . 3 |- ((ph /\ x e. C) -> E!y y = B)
6		eleq1 1957	. . . . . . 7 |- (y = B -> (y e. C <-> B e. C))
7	6, 1	syl5cbir 228	. . . . . 6 |- ((ph /\ x e. C) -> (y = B -> y e. C))
8	7	pm4.71rd 701	. . . . 5 |- ((ph /\ x e. C) -> (y = B <-> (y e. C /\ y = B)))
9		reuhypd.2	. . . . . . 7 |- ((ph /\ x e. C /\ y e. C) -> (x = A <-> y = B))
10	9	3expa 1067	. . . . . 6 |- (((ph /\ x e. C) /\ y e. C) -> (x = A <-> y = B))
11	10	pm5.32da 711	. . . . 5 |- ((ph /\ x e. C) -> ((y e. C /\ x = A) <-> (y e. C /\ y = B)))
12	8, 11	bitr4d 590	. . . 4 |- ((ph /\ x e. C) -> (y = B <-> (y e. C /\ x = A)))
13	12	eubidv 1779	. . 3 |- ((ph /\ x e. C) -> (E!y y = B <-> E!y(y e. C /\ x = A)))
14	5, 13	mpbid 212	. 2 |- ((ph /\ x e. C) -> E!y(y e. C /\ x = A))
15		df-reu 2111	. 2 |- (E!y e. C x = A <-> E!y(y e. C /\ x = A))
16	14, 15	sylibr 217	1 |- ((ph /\ x e. C) -> E!y e. C x = A)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  E!weu 1771  E!wreu 2107  _Vcvv 2292

This theorem is referenced by:  reuhyp 3849  riotaoc 16936

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-reu 2111  df-v 2294

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