Theorem reuhyp 4222

Description: A theorem useful for eliminating restricted existential uniqueness hypotheses. (Contributed by NM, 15-Nov-2004.)

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

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

Proof of Theorem reuhyp

Step	Hyp	Ref	Expression
1		tru 1288	. 2 |- T.
2		reuhyp.1	. . . 4 |- ( x e. C -> B e. C )
3	2	adantl 271	. . 3 |- ( ( T. /\ x e. C ) -> B e. C )
4		reuhyp.2	. . . 4 |- ( ( x e. C /\ y e. C ) -> ( x = A <-> y = B ) )
5	4	3adant1 956	. . 3 |- ( ( T. /\ x e. C /\ y e. C ) -> ( x = A <-> y = B ) )
6	3, 5	reuhypd 4221	. 2 |- ( ( T. /\ x e. C ) -> E! y e. C x = A )
7	1, 6	mpan 414	1 |- ( x e. C -> E! y e. C x = A )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   T. wtru 1285   e. wcel 1433   E!wreu 2350

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-reu 2355  df-v 2603

This theorem is referenced by: (None)