Theorem euxfrdc 2721

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
euxfrdc.1	|- A e. _V
euxfrdc.2	|- E!y x = A
euxfrdc.3	|- (x = A -> (ph <-> ps))

Assertion

Ref	Expression
euxfrdc	|- (DECID E.yE.x(x = A /\ ps) -> (E!xph <-> E!yps))

Distinct variable groups:   ps,x   ph,y   x,A

Allowed substitution hints:   ph(x)   ps(y)   A(y)

Proof of Theorem euxfrdc

Step	Hyp	Ref	Expression
1		euxfrdc.2	. . . . . 6 |- E!y x = A
2		euex 1927	. . . . . 6 |- (E!y x = A -> E.y x = A)
3	1, 2	ax-mp 7	. . . . 5 |- E.y x = A
4	3	biantrur 287	. . . 4 |- (ph <-> (E.y x = A /\ ph))
5		19.41v 1779	. . . 4 |- (E.y(x = A /\ ph) <-> (E.y x = A /\ ph))
6		euxfrdc.3	. . . . . 6 |- (x = A -> (ph <-> ps))
7	6	pm5.32i 427	. . . . 5 |- ((x = A /\ ph) <-> (x = A /\ ps))
8	7	exbii 1493	. . . 4 |- (E.y(x = A /\ ph) <-> E.y(x = A /\ ps))
9	4, 5, 8	3bitr2i 197	. . 3 |- (ph <-> E.y(x = A /\ ps))
10	9	eubii 1906	. 2 |- (E!xph <-> E!xE.y(x = A /\ ps))
11		euxfrdc.1	. . 3 |- A e. _V
12	1	eumoi 1930	. . 3 |- E*y x = A
13	11, 12	euxfr2dc 2720	. 2 |- (DECID E.yE.x(x = A /\ ps) -> (E!xE.y(x = A /\ ps) <-> E!yps))
14	10, 13	syl5bb 181	1 |- (DECID E.yE.x(x = A /\ ps) -> (E!xph <-> E!yps))

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98  DECID wdc 741   = wceq 1242  E.wex 1378   e. wcel 1390  E!weu 1897   _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-bndl 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: (None)