Theorem reuhypd 4203

Description: A theorem useful for eliminating restricted existential uniqueness hypotheses. (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		elex 2566	. . . . 5 |- ( B e. C -> B e. _V )
3	1, 2	syl 14	. . . 4 |- ( ( ph /\ x e. C ) -> B e. _V )
4		eueq 2712	. . . 4 |- ( B e. _V <-> E! y y = B )
5	3, 4	sylib 127	. . 3 |- ( ( ph /\ x e. C ) -> E! y y = B )
6		eleq1 2100	. . . . . . 7 |- ( y = B -> ( y e. C <-> B e. C ) )
7	1, 6	syl5ibrcom 146	. . . . . 6 |- ( ( ph /\ x e. C ) -> ( y = B -> y e. C ) )
8	7	pm4.71rd 374	. . . . 5 |- ( ( ph /\ x e. C ) -> ( y = B <-> ( y e. C /\ y = B ) ) )
9		reuhypd.2	. . . . . . 7 |- ( ( ph /\ x e. C /\ y e. C ) -> ( x = A <-> y = B ) )
10	9	3expa 1104	. . . . . 6 |- ( ( ( ph /\ x e. C ) /\ y e. C ) -> ( x = A <-> y = B ) )
11	10	pm5.32da 425	. . . . 5 |- ( ( ph /\ x e. C ) -> ( ( y e. C /\ x = A ) <-> ( y e. C /\ y = B ) ) )
12	8, 11	bitr4d 180	. . . 4 |- ( ( ph /\ x e. C ) -> ( y = B <-> ( y e. C /\ x = A ) ) )
13	12	eubidv 1908	. . 3 |- ( ( ph /\ x e. C ) -> ( E! y y = B <-> E! y ( y e. C /\ x = A ) ) )
14	5, 13	mpbid 135	. 2 |- ( ( ph /\ x e. C ) -> E! y ( y e. C /\ x = A ) )
15		df-reu 2313	. 2 |- ( E! y e. C x = A <-> E! y ( y e. C /\ x = A ) )
16	14, 15	sylibr 137	1 |- ( ( ph /\ x e. C ) -> E! y e. C x = A )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   /\ w3a 885   = wceq 1243   e. wcel 1393   E!weu 1900   E!wreu 2308   _Vcvv 2557

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-reu 2313  df-v 2559

This theorem is referenced by:  reuhyp  4204