Theorem reuhypd 4169

Description: A theorem useful for eliminating restricted existential uniqueness hypotheses. (Contributed by NM, 16-Jan-2012.)

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

Allowed substitution hints:   ph(x)   A(x,y)   B(x)   C(x)

Proof of Theorem reuhypd

Step	Hyp	Ref	Expression
1		reuhypd.1	. . . . 5 |- ((ph /\ x e. C) -> B e. C)
2		elex 2560	. . . . 5 |- (B e. C -> B e. _V)
3	1, 2	syl 14	. . . 4 |- ((ph /\ x e. C) -> B e. _V)
4		eueq 2706	. . . 4 |- (B e. _V <-> E!y y = B)
5	3, 4	sylib 127	. . 3 |- ((ph /\ x e. C) -> E!y y = B)
6		eleq1 2097	. . . . . . 7 |- (y = B -> (y e. C <-> B e. C))
7	1, 6	syl5ibrcom 146	. . . . . 6 |- ((ph /\ x e. C) -> (y = B -> y e. C))
8	7	pm4.71rd 374	. . . . 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 1103	. . . . . 6 |- (((ph /\ x e. C) /\ y e. C) -> (x = A <-> y = B))
11	10	pm5.32da 425	. . . . 5 |- ((ph /\ x e. C) -> ((y e. C /\ x = A) <-> (y e. C /\ y = B)))
12	8, 11	bitr4d 180	. . . 4 |- ((ph /\ x e. C) -> (y = B <-> (y e. C /\ x = A)))
13	12	eubidv 1905	. . 3 |- ((ph /\ x e. C) -> (E!y y = B <-> E!y(y e. C /\ x = A)))
14	5, 13	mpbid 135	. 2 |- ((ph /\ x e. C) -> E!y(y e. C /\ x = A))
15		df-reu 2307	. 2 |- (E!y e. C x = A <-> E!y(y e. C /\ x = A))
16	14, 15	sylibr 137	1 |- ((ph /\ x e. C) -> E!y e. C x = A)

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   /\ w3a 884   = wceq 1242   e. wcel 1390  E!weu 1897  E!wreu 2302   _Vcvv 2551

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-3an 886  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-reu 2307  df-v 2553

This theorem is referenced by:  reuhyp  4170