Theorem reuhypd 4449

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 2737	. . . . 5 |- ( B e. C -> B e. _V )
3	1, 2	syl 14	. . . 4 |- ( ( ph /\ x e. C ) -> B e. _V )
4		eueq 2897	. . . 4 |- ( B e. _V <-> E! y y = B )
5	3, 4	sylib 121	. . 3 |- ( ( ph /\ x e. C ) -> E! y y = B )
6		eleq1 2229	. . . . . . 7 |- ( y = B -> ( y e. C <-> B e. C ) )
7	1, 6	syl5ibrcom 156	. . . . . 6 |- ( ( ph /\ x e. C ) -> ( y = B -> y e. C ) )
8	7	pm4.71rd 392	. . . . 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 1193	. . . . . 6 |- ( ( ( ph /\ x e. C ) /\ y e. C ) -> ( x = A <-> y = B ) )
11	10	pm5.32da 448	. . . . 5 |- ( ( ph /\ x e. C ) -> ( ( y e. C /\ x = A ) <-> ( y e. C /\ y = B ) ) )
12	8, 11	bitr4d 190	. . . 4 |- ( ( ph /\ x e. C ) -> ( y = B <-> ( y e. C /\ x = A ) ) )
13	12	eubidv 2022	. . 3 |- ( ( ph /\ x e. C ) -> ( E! y y = B <-> E! y ( y e. C /\ x = A ) ) )
14	5, 13	mpbid 146	. 2 |- ( ( ph /\ x e. C ) -> E! y ( y e. C /\ x = A ) )
15		df-reu 2451	. 2 |- ( E! y e. C x = A <-> E! y ( y e. C /\ x = A ) )
16	14, 15	sylibr 133	1 |- ( ( ph /\ x e. C ) -> E! y e. C x = A )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 968   = wceq 1343   E!weu 2014   e. wcel 2136   E!wreu 2446   _Vcvv 2726

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-reu 2451  df-v 2728

This theorem is referenced by:  reuhyp  4450