Theorem euxfr2dc 2777

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
euxfr2dc.1	|- A e. _V
euxfr2dc.2	|- E* y x = A

Assertion

Ref	Expression
euxfr2dc	|- (DECID E. y E. x ( x = A /\ ph ) -> ( E! x E. y ( x = A /\ ph ) <-> E! y ph ) )

Distinct variable groups:   ph, x   x, A

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

Proof of Theorem euxfr2dc

Step	Hyp	Ref	Expression
1		euxfr2dc.2	. . . . . . 7 |- E* y x = A
2	1	moani 2011	. . . . . 6 |- E* y ( ph /\ x = A )
3		ancom 262	. . . . . . 7 |- ( ( ph /\ x = A ) <-> ( x = A /\ ph ) )
4	3	mobii 1978	. . . . . 6 |- ( E* y ( ph /\ x = A ) <-> E* y ( x = A /\ ph ) )
5	2, 4	mpbi 143	. . . . 5 |- E* y ( x = A /\ ph )
6	5	ax-gen 1378	. . . 4 |- A. x E* y ( x = A /\ ph )
7		excom 1594	. . . . . 6 |- ( E. y E. x ( x = A /\ ph ) <-> E. x E. y ( x = A /\ ph ) )
8	7	dcbii 780	. . . . 5 |- (DECID E. y E. x ( x = A /\ ph ) <-> DECID E. x E. y ( x = A /\ ph ) )
9		2euswapdc 2032	. . . . 5 |- (DECID E. x E. y ( x = A /\ ph ) -> ( A. x E* y ( x = A /\ ph ) -> ( E! x E. y ( x = A /\ ph ) -> E! y E. x ( x = A /\ ph ) ) ) )
10	8, 9	sylbi 119	. . . 4 |- (DECID E. y E. x ( x = A /\ ph ) -> ( A. x E* y ( x = A /\ ph ) -> ( E! x E. y ( x = A /\ ph ) -> E! y E. x ( x = A /\ ph ) ) ) )
11	6, 10	mpi 15	. . 3 |- (DECID E. y E. x ( x = A /\ ph ) -> ( E! x E. y ( x = A /\ ph ) -> E! y E. x ( x = A /\ ph ) ) )
12		moeq 2767	. . . . . . 7 |- E* x x = A
13	12	moani 2011	. . . . . 6 |- E* x ( ph /\ x = A )
14	3	mobii 1978	. . . . . 6 |- ( E* x ( ph /\ x = A ) <-> E* x ( x = A /\ ph ) )
15	13, 14	mpbi 143	. . . . 5 |- E* x ( x = A /\ ph )
16	15	ax-gen 1378	. . . 4 |- A. y E* x ( x = A /\ ph )
17		2euswapdc 2032	. . . 4 |- (DECID E. y E. x ( x = A /\ ph ) -> ( A. y E* x ( x = A /\ ph ) -> ( E! y E. x ( x = A /\ ph ) -> E! x E. y ( x = A /\ ph ) ) ) )
18	16, 17	mpi 15	. . 3 |- (DECID E. y E. x ( x = A /\ ph ) -> ( E! y E. x ( x = A /\ ph ) -> E! x E. y ( x = A /\ ph ) ) )
19	11, 18	impbid 127	. 2 |- (DECID E. y E. x ( x = A /\ ph ) -> ( E! x E. y ( x = A /\ ph ) <-> E! y E. x ( x = A /\ ph ) ) )
20		euxfr2dc.1	. . . 4 |- A e. _V
21		biidd 170	. . . 4 |- ( x = A -> ( ph <-> ph ) )
22	20, 21	ceqsexv 2638	. . 3 |- ( E. x ( x = A /\ ph ) <-> ph )
23	22	eubii 1950	. 2 |- ( E! y E. x ( x = A /\ ph ) <-> E! y ph )
24	19, 23	syl6bb 194	1 |- (DECID E. y E. x ( x = A /\ ph ) -> ( E! x E. y ( x = A /\ ph ) <-> E! y ph ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103  DECID wdc 775   A.wal 1282   = wceq 1284   E.wex 1421   e. wcel 1433   E!weu 1941   E*wmo 1942   _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:  euxfrdc  2778