Theorem reuhyp 4388

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 1335	. 2 |- T.
2		reuhyp.1	. . . 4 |- ( x e. C -> B e. C )
3	2	adantl 275	. . 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 999	. . 3 |- ( ( T. /\ x e. C /\ y e. C ) -> ( x = A <-> y = B ) )
6	3, 5	reuhypd 4387	. 2 |- ( ( T. /\ x e. C ) -> E! y e. C x = A )
7	1, 6	mpan 420	1 |- ( x e. C -> E! y e. C x = A )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   T. wtru 1332   e. wcel 1480   E!wreu 2416

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-reu 2421  df-v 2683

This theorem is referenced by: (None)