Theorem euxfr2dc 2726

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 1970	. . . . . 6 |- E* y ( ph /\ x = A )
3		ancom 253	. . . . . . 7 |- ( ( ph /\ x = A ) <-> ( x = A /\ ph ) )
4	3	mobii 1937	. . . . . 6 |- ( E* y ( ph /\ x = A ) <-> E* y ( x = A /\ ph ) )
5	2, 4	mpbi 133	. . . . 5 |- E* y ( x = A /\ ph )
6	5	ax-gen 1338	. . . 4 |- A. x E* y ( x = A /\ ph )
7		excom 1554	. . . . . 6 |- ( E. y E. x ( x = A /\ ph ) <-> E. x E. y ( x = A /\ ph ) )
8	7	dcbii 747	. . . . 5 |- (DECID E. y E. x ( x = A /\ ph ) <-> DECID E. x E. y ( x = A /\ ph ) )
9		2euswapdc 1991	. . . . 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 114	. . . 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 2716	. . . . . . 7 |- E* x x = A
13	12	moani 1970	. . . . . 6 |- E* x ( ph /\ x = A )
14	3	mobii 1937	. . . . . 6 |- ( E* x ( ph /\ x = A ) <-> E* x ( x = A /\ ph ) )
15	13, 14	mpbi 133	. . . . 5 |- E* x ( x = A /\ ph )
16	15	ax-gen 1338	. . . 4 |- A. y E* x ( x = A /\ ph )
17		2euswapdc 1991	. . . 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 120	. 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 161	. . . 4 |- ( x = A -> ( ph <-> ph ) )
22	20, 21	ceqsexv 2593	. . 3 |- ( E. x ( x = A /\ ph ) <-> ph )
23	22	eubii 1909	. 2 |- ( E! y E. x ( x = A /\ ph ) <-> E! y ph )
24	19, 23	syl6bb 185	1 |- (DECID E. y E. x ( x = A /\ ph ) -> ( E! x E. y ( x = A /\ ph ) <-> E! y ph ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98  DECID wdc 742   A.wal 1241   = wceq 1243   E.wex 1381   e. wcel 1393   E!weu 1900   E*wmo 1901   _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-in1 544  ax-in2 545  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-dc 743  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-v 2559

This theorem is referenced by:  euxfrdc  2727