Theorem reuhyp 4466

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 1357	. 2 |- T.
2		reuhyp.1	. . . 4 |- ( x e. C -> B e. C )
3	2	adantl 277	. . 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 1015	. . 3 |- ( ( T. /\ x e. C /\ y e. C ) -> ( x = A <-> y = B ) )
6	3, 5	reuhypd 4465	. 2 |- ( ( T. /\ x e. C ) -> E! y e. C x = A )
7	1, 6	mpan 424	1 |- ( x e. C -> E! y e. C x = A )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   T. wtru 1354   e. wcel 2146   E!wreu 2455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-reu 2460  df-v 2737

This theorem is referenced by: (None)