Theorem reuhyp 4562

Description: A theorem useful for eliminating the restricted existential uniqueness hypotheses in reuxfr 4560. (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 1312	. 2 |- T.
2		reuhyp.1	. . . 4 |- ( x e. C -> B e. C )
3	2	adantl 452	. . 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 973	. . 3 |- ( ( T. /\ x e. C /\ y e. C ) -> ( x = A <-> y = B ) )
6	3, 5	reuhypd 4561	. 2 |- ( ( T. /\ x e. C ) -> E! y e. C x = A )
7	1, 6	mpan 651	1 |- ( x e. C -> E! y e. C x = A )

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   T. wtru 1307   = wceq 1623   e. wcel 1684   E!wreu 2545

This theorem is referenced by:  riotaneg  9729  zmax  10313  rebtwnz  10315

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-reu 2550  df-v 2790