Theorem reuhyp 4718

Description: A theorem useful for eliminating the restricted existential uniqueness hypotheses in reuxfr 4716. (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 1327	. 2 |- T.
2		reuhyp.1	. . . 4 |- ( x e. C -> B e. C )
3	2	adantl 453	. . 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 975	. . 3 |- ( ( T. /\ x e. C /\ y e. C ) -> ( x = A <-> y = B ) )
6	3, 5	reuhypd 4717	. 2 |- ( ( T. /\ x e. C ) -> E! y e. C x = A )
7	1, 6	mpan 652	1 |- ( x e. C -> E! y e. C x = A )

Syntax hints:   -> wi 4   <-> wb 177   /\ wa 359   T. wtru 1322   = wceq 1649   e. wcel 1721   E!wreu 2676

This theorem is referenced by:  riotaneg  9947  zmax  10535  rebtwnz  10537

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393

This theorem depends on definitions:  df-bi 178  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-reu 2681  df-v 2926