Theorem euxfr2dc 2720

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.yE.x(x = A /\ ph) -> (E!xE.y(x = A /\ ph) <-> E!yph))

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 1967	. . . . . 6 |- E*y(ph /\ x = A)
3		ancom 253	. . . . . . 7 |- ((ph /\ x = A) <-> (x = A /\ ph))
4	3	mobii 1934	. . . . . 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 1335	. . . 4 |- A.xE*y(x = A /\ ph)
7		excom 1551	. . . . . 6 |- (E.yE.x(x = A /\ ph) <-> E.xE.y(x = A /\ ph))
8	7	dcbii 746	. . . . 5 |- (DECID E.yE.x(x = A /\ ph) <-> DECID E.xE.y(x = A /\ ph))
9		2euswapdc 1988	. . . . 5 |- (DECID E.xE.y(x = A /\ ph) -> (A.xE*y(x = A /\ ph) -> (E!xE.y(x = A /\ ph) -> E!yE.x(x = A /\ ph))))
10	8, 9	sylbi 114	. . . 4 |- (DECID E.yE.x(x = A /\ ph) -> (A.xE*y(x = A /\ ph) -> (E!xE.y(x = A /\ ph) -> E!yE.x(x = A /\ ph))))
11	6, 10	mpi 15	. . 3 |- (DECID E.yE.x(x = A /\ ph) -> (E!xE.y(x = A /\ ph) -> E!yE.x(x = A /\ ph)))
12		moeq 2710	. . . . . . 7 |- E*x x = A
13	12	moani 1967	. . . . . 6 |- E*x(ph /\ x = A)
14	3	mobii 1934	. . . . . 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 1335	. . . 4 |- A.yE*x(x = A /\ ph)
17		2euswapdc 1988	. . . 4 |- (DECID E.yE.x(x = A /\ ph) -> (A.yE*x(x = A /\ ph) -> (E!yE.x(x = A /\ ph) -> E!xE.y(x = A /\ ph))))
18	16, 17	mpi 15	. . 3 |- (DECID E.yE.x(x = A /\ ph) -> (E!yE.x(x = A /\ ph) -> E!xE.y(x = A /\ ph)))
19	11, 18	impbid 120	. 2 |- (DECID E.yE.x(x = A /\ ph) -> (E!xE.y(x = A /\ ph) <-> E!yE.x(x = A /\ ph)))
20		euxfr2dc.1	. . . 4 |- A e. _V
21		biidd 161	. . . 4 |- (x = A -> (ph <-> ph))
22	20, 21	ceqsexv 2587	. . 3 |- (E.x(x = A /\ ph) <-> ph)
23	22	eubii 1906	. 2 |- (E!yE.x(x = A /\ ph) <-> E!yph)
24	19, 23	syl6bb 185	1 |- (DECID E.yE.x(x = A /\ ph) -> (E!xE.y(x = A /\ ph) <-> E!yph))

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98  DECID wdc 741  A.wal 1240   = wceq 1242  E.wex 1378   e. wcel 1390  E!weu 1897  E*wmo 1898   _Vcvv 2551

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-dc 742  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-v 2553

This theorem is referenced by:  euxfrdc  2721