Theorem euxfrdc 2778

Description: Transfer existential uniqueness from a variable x to another variable y contained in expression A. (Contributed by NM, 14-Nov-2004.)

Hypotheses

Ref	Expression
euxfrdc.1	|- A e. _V
euxfrdc.2	|- E! y x = A
euxfrdc.3	|- ( x = A -> ( ph <-> ps ) )

Assertion

Ref	Expression
euxfrdc	|- (DECID E. y E. x ( x = A /\ ps ) -> ( E! x ph <-> E! y ps ) )

Distinct variable groups:   ps, x   ph, y   x, A

Allowed substitution hints:   ph( x)   ps( y)   A( y)

Proof of Theorem euxfrdc

Step	Hyp	Ref	Expression
1		euxfrdc.2	. . . . . 6 |- E! y x = A
2		euex 1971	. . . . . 6 |- ( E! y x = A -> E. y x = A )
3	1, 2	ax-mp 7	. . . . 5 |- E. y x = A
4	3	biantrur 297	. . . 4 |- ( ph <-> ( E. y x = A /\ ph ) )
5		19.41v 1823	. . . 4 |- ( E. y ( x = A /\ ph ) <-> ( E. y x = A /\ ph ) )
6		euxfrdc.3	. . . . . 6 |- ( x = A -> ( ph <-> ps ) )
7	6	pm5.32i 441	. . . . 5 |- ( ( x = A /\ ph ) <-> ( x = A /\ ps ) )
8	7	exbii 1536	. . . 4 |- ( E. y ( x = A /\ ph ) <-> E. y ( x = A /\ ps ) )
9	4, 5, 8	3bitr2i 206	. . 3 |- ( ph <-> E. y ( x = A /\ ps ) )
10	9	eubii 1950	. 2 |- ( E! x ph <-> E! x E. y ( x = A /\ ps ) )
11		euxfrdc.1	. . 3 |- A e. _V
12	1	eumoi 1974	. . 3 |- E* y x = A
13	11, 12	euxfr2dc 2777	. 2 |- (DECID E. y E. x ( x = A /\ ps ) -> ( E! x E. y ( x = A /\ ps ) <-> E! y ps ) )
14	10, 13	syl5bb 190	1 |- (DECID E. y E. x ( x = A /\ ps ) -> ( E! x ph <-> E! y ps ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103  DECID wdc 775   = wceq 1284   E.wex 1421   e. wcel 1433   E!weu 1941   _Vcvv 2601

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-dc 776  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-v 2603

This theorem is referenced by: (None)