Theorem euxfr2dc 2991

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 2150	. . . . . 6 |- E* y ( ph /\ x = A )
3		ancom 266	. . . . . . 7 |- ( ( ph /\ x = A ) <-> ( x = A /\ ph ) )
4	3	mobii 2116	. . . . . 6 |- ( E* y ( ph /\ x = A ) <-> E* y ( x = A /\ ph ) )
5	2, 4	mpbi 145	. . . . 5 |- E* y ( x = A /\ ph )
6	5	ax-gen 1497	. . . 4 |- A. x E* y ( x = A /\ ph )
7		excom 1712	. . . . . 6 |- ( E. y E. x ( x = A /\ ph ) <-> E. x E. y ( x = A /\ ph ) )
8	7	dcbii 847	. . . . 5 |- (DECID E. y E. x ( x = A /\ ph ) <-> DECID E. x E. y ( x = A /\ ph ) )
9		2euswapdc 2171	. . . . 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 121	. . . 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 2981	. . . . . . 7 |- E* x x = A
13	12	moani 2150	. . . . . 6 |- E* x ( ph /\ x = A )
14	3	mobii 2116	. . . . . 6 |- ( E* x ( ph /\ x = A ) <-> E* x ( x = A /\ ph ) )
15	13, 14	mpbi 145	. . . . 5 |- E* x ( x = A /\ ph )
16	15	ax-gen 1497	. . . 4 |- A. y E* x ( x = A /\ ph )
17		2euswapdc 2171	. . . 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 129	. 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 172	. . . 4 |- ( x = A -> ( ph <-> ph ) )
22	20, 21	ceqsexv 2842	. . 3 |- ( E. x ( x = A /\ ph ) <-> ph )
23	22	eubii 2088	. 2 |- ( E! y E. x ( x = A /\ ph ) <-> E! y ph )
24	19, 23	bitrdi 196	1 |- (DECID E. y E. x ( x = A /\ ph ) -> ( E! x E. y ( x = A /\ ph ) <-> E! y ph ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 841   A.wal 1395   = wceq 1397   E.wex 1540   E!weu 2079   E*wmo 2080   e. wcel 2202   _Vcvv 2802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-dc 842  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-v 2804

This theorem is referenced by:  euxfrdc  2992