Theorem euxfr2dc 2838

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 2045	. . . . . 6 |- E* y ( ph /\ x = A )
3		ancom 264	. . . . . . 7 |- ( ( ph /\ x = A ) <-> ( x = A /\ ph ) )
4	3	mobii 2012	. . . . . 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 1408	. . . 4 |- A. x E* y ( x = A /\ ph )
7		excom 1625	. . . . . 6 |- ( E. y E. x ( x = A /\ ph ) <-> E. x E. y ( x = A /\ ph ) )
8	7	dcbii 808	. . . . 5 |- (DECID E. y E. x ( x = A /\ ph ) <-> DECID E. x E. y ( x = A /\ ph ) )
9		2euswapdc 2066	. . . . 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 2828	. . . . . . 7 |- E* x x = A
13	12	moani 2045	. . . . . 6 |- E* x ( ph /\ x = A )
14	3	mobii 2012	. . . . . 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 1408	. . . 4 |- A. y E* x ( x = A /\ ph )
17		2euswapdc 2066	. . . 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 2696	. . 3 |- ( E. x ( x = A /\ ph ) <-> ph )
23	22	eubii 1984	. 2 |- ( E! y E. x ( x = A /\ ph ) <-> E! y ph )
24	19, 23	syl6bb 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 802   A.wal 1312   = wceq 1314   E.wex 1451   e. wcel 1463   E!weu 1975   E*wmo 1976   _Vcvv 2657

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-dc 803  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-v 2659

This theorem is referenced by:  euxfrdc  2839