Theorem reuhypd 4895

Description: A theorem useful for eliminating the restricted existential uniqueness hypotheses in riotaxfrd 6642. (Contributed by NM, 16-Jan-2012.)

Hypotheses

Ref	Expression
reuhypd.1	|- ( ( ph /\ x e. C ) -> B e. C )
reuhypd.2	|- ( ( ph /\ x e. C /\ y e. C ) -> ( x = A <-> y = B ) )

Assertion

Ref	Expression
reuhypd	|- ( ( ph /\ x e. C ) -> E! y e. C x = A )

Distinct variable groups:   ph, y   y, B   y, C   x, y

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

Proof of Theorem reuhypd

Step	Hyp	Ref	Expression
1		reuhypd.1	. . . . 5 |- ( ( ph /\ x e. C ) -> B e. C )
2	1	elexd 3214	. . . 4 |- ( ( ph /\ x e. C ) -> B e. _V )
3		eueq 3378	. . . 4 |- ( B e. _V <-> E! y y = B )
4	2, 3	sylib 208	. . 3 |- ( ( ph /\ x e. C ) -> E! y y = B )
5		eleq1 2689	. . . . . . 7 |- ( y = B -> ( y e. C <-> B e. C ) )
6	1, 5	syl5ibrcom 237	. . . . . 6 |- ( ( ph /\ x e. C ) -> ( y = B -> y e. C ) )
7	6	pm4.71rd 667	. . . . 5 |- ( ( ph /\ x e. C ) -> ( y = B <-> ( y e. C /\ y = B ) ) )
8		reuhypd.2	. . . . . . 7 |- ( ( ph /\ x e. C /\ y e. C ) -> ( x = A <-> y = B ) )
9	8	3expa 1265	. . . . . 6 |- ( ( ( ph /\ x e. C ) /\ y e. C ) -> ( x = A <-> y = B ) )
10	9	pm5.32da 673	. . . . 5 |- ( ( ph /\ x e. C ) -> ( ( y e. C /\ x = A ) <-> ( y e. C /\ y = B ) ) )
11	7, 10	bitr4d 271	. . . 4 |- ( ( ph /\ x e. C ) -> ( y = B <-> ( y e. C /\ x = A ) ) )
12	11	eubidv 2490	. . 3 |- ( ( ph /\ x e. C ) -> ( E! y y = B <-> E! y ( y e. C /\ x = A ) ) )
13	4, 12	mpbid 222	. 2 |- ( ( ph /\ x e. C ) -> E! y ( y e. C /\ x = A ) )
14		df-reu 2919	. 2 |- ( E! y e. C x = A <-> E! y ( y e. C /\ x = A ) )
15	13, 14	sylibr 224	1 |- ( ( ph /\ x e. C ) -> E! y e. C x = A )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E!weu 2470   E!wreu 2914   _Vcvv 3200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-reu 2919  df-v 3202

This theorem is referenced by:  reuhyp  4896  riotaocN  34496