Theorem euxfr2dc 2911

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 2084	. . . . . 6 |- E* y ( ph /\ x = A )
3		ancom 264	. . . . . . 7 |- ( ( ph /\ x = A ) <-> ( x = A /\ ph ) )
4	3	mobii 2051	. . . . . 6 |- ( E* y ( ph /\ x = A ) <-> E* y ( x = A /\ ph ) )
5	2, 4	mpbi 144	. . . . 5 |- E* y ( x = A /\ ph )
6	5	ax-gen 1437	. . . 4 |- A. x E* y ( x = A /\ ph )
7		excom 1652	. . . . . 6 |- ( E. y E. x ( x = A /\ ph ) <-> E. x E. y ( x = A /\ ph ) )
8	7	dcbii 830	. . . . 5 |- (DECID E. y E. x ( x = A /\ ph ) <-> DECID E. x E. y ( x = A /\ ph ) )
9		2euswapdc 2105	. . . . 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 120	. . . 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 2901	. . . . . . 7 |- E* x x = A
13	12	moani 2084	. . . . . 6 |- E* x ( ph /\ x = A )
14	3	mobii 2051	. . . . . 6 |- ( E* x ( ph /\ x = A ) <-> E* x ( x = A /\ ph ) )
15	13, 14	mpbi 144	. . . . 5 |- E* x ( x = A /\ ph )
16	15	ax-gen 1437	. . . 4 |- A. y E* x ( x = A /\ ph )
17		2euswapdc 2105	. . . 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 128	. 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 171	. . . 4 |- ( x = A -> ( ph <-> ph ) )
22	20, 21	ceqsexv 2765	. . 3 |- ( E. x ( x = A /\ ph ) <-> ph )
23	22	eubii 2023	. 2 |- ( E! y E. x ( x = A /\ ph ) <-> E! y ph )
24	19, 23	bitrdi 195	1 |- (DECID E. y E. x ( x = A /\ ph ) -> ( E! x E. y ( x = A /\ ph ) <-> E! y ph ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104  DECID wdc 824   A.wal 1341   = wceq 1343   E.wex 1480   E!weu 2014   E*wmo 2015   e. wcel 2136   _Vcvv 2726

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-dc 825  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-v 2728

This theorem is referenced by:  euxfrdc  2912