Theorem euxfr2 1926

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 1445	. . . 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 1423	. . . . 5 |- E*y(ph /\ x = A)
4		ancom 435	. . . . . 6 |- ((ph /\ x = A) <-> (x = A /\ ph))
5	4	mobii 1405	. . . . 5 |- (E*y(ph /\ x = A) <-> E*y(x = A /\ ph))
6	3, 5	mpbi 189	. . . 4 |- E*y(x = A /\ ph)
7	1, 6	mpg 986	. . 3 |- (E!xE.y(x = A /\ ph) -> E!yE.x(x = A /\ ph))
8		2euswap 1445	. . . 4 |- (A.yE*x(x = A /\ ph) -> (E!yE.x(x = A /\ ph) -> E!xE.y(x = A /\ ph)))
9		moeq 1920	. . . . . 6 |- E*x x = A
10	9	moani 1423	. . . . 5 |- E*x(ph /\ x = A)
11	4	mobii 1405	. . . . 5 |- (E*x(ph /\ x = A) <-> E*x(x = A /\ ph))
12	10, 11	mpbi 189	. . . 4 |- E*x(x = A /\ ph)
13	8, 12	mpg 986	. . 3 |- (E!yE.x(x = A /\ ph) -> E!xE.y(x = A /\ ph))
14	7, 13	impbi 157	. 2 |- (E!xE.y(x = A /\ ph) <-> E!yE.x(x = A /\ ph))
15		euxfr2.1	. . . 4 |- A e. V
16		pm4.2d 171	. . . 4 |- (x = A -> (ph <-> ph))
17	15, 16	ceqsexv 1835	. . 3 |- (E.x(x = A /\ ph) <-> ph)
18	17	eubii 1387	. 2 |- (E!yE.x(x = A /\ ph) <-> E!yph)
19	14, 18	bitr 173	1 |- (E!xE.y(x = A /\ ph) <-> E!yph)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  E.wex 980  E!weu 1380  E*wmo 1381  Vcvv 1811

This theorem is referenced by:  euxfr 1927  euop2 2806

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812

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