Theorem euxfr2 2437

Description: Transfer existential uniqueness from a variable x to another variable y contained in expression A.

Hypotheses

Ref	Expression
euxfr2.1	|- A e. _V
euxfr2.2	|- E*y x = A

Assertion

Ref	Expression
euxfr2	|- (E!xE.y(x = A /\ ph) <-> E!yph)

Distinct variable groups:   ph,x   x,A

Proof of Theorem euxfr2

Step	Hyp	Ref	Expression
1		2euswap 1849	. . . 4 |- (A.xE*y(x = A /\ ph) -> (E!xE.y(x = A /\ ph) -> E!yE.x(x = A /\ ph)))
2		euxfr2.2	. . . . . 6 |- E*y x = A
3	2	moani 1820	. . . . 5 |- E*y(ph /\ x = A)
4		ancom 482	. . . . . 6 |- ((ph /\ x = A) <-> (x = A /\ ph))
5	4	mobii 1801	. . . . 5 |- (E*y(ph /\ x = A) <-> E*y(x = A /\ ph))
6	3, 5	mpbi 206	. . . 4 |- E*y(x = A /\ ph)
7	1, 6	mpg 1332	. . 3 |- (E!xE.y(x = A /\ ph) -> E!yE.x(x = A /\ ph))
8		2euswap 1849	. . . 4 |- (A.yE*x(x = A /\ ph) -> (E!yE.x(x = A /\ ph) -> E!xE.y(x = A /\ ph)))
9		moeq 2431	. . . . . 6 |- E*x x = A
10	9	moani 1820	. . . . 5 |- E*x(ph /\ x = A)
11	4	mobii 1801	. . . . 5 |- (E*x(ph /\ x = A) <-> E*x(x = A /\ ph))
12	10, 11	mpbi 206	. . . 4 |- E*x(x = A /\ ph)
13	8, 12	mpg 1332	. . 3 |- (E!yE.x(x = A /\ ph) -> E!xE.y(x = A /\ ph))
14	7, 13	impbii 174	. 2 |- (E!xE.y(x = A /\ ph) <-> E!yE.x(x = A /\ ph))
15		euxfr2.1	. . . 4 |- A e. _V
16		biidd 188	. . . 4 |- (x = A -> (ph <-> ph))
17	15, 16	ceqsexv 2325	. . 3 |- (E.x(x = A /\ ph) <-> ph)
18	17	eubii 1780	. 2 |- (E!yE.x(x = A /\ ph) <-> E!yph)
19	14, 18	bitri 190	1 |- (E!xE.y(x = A /\ ph) <-> E!yph)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  E!weu 1771  E*wmo 1772  _Vcvv 2292

This theorem is referenced by:  euxfr 2438  euop2 3553

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294

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