Theorem euxfr2 3022

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
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

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

Proof of Theorem euxfr2

Step	Hyp	Ref	Expression
1		2euswap 2280	. . . 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 2256	. . . . 5 |- E*y(ph /\ x = A)
4		ancom 437	. . . . . 6 |- ((ph /\ x = A) <-> (x = A /\ ph))
5	4	mobii 2240	. . . . 5 |- (E*y(ph /\ x = A) <-> E*y(x = A /\ ph))
6	3, 5	mpbi 199	. . . 4 |- E*y(x = A /\ ph)
7	1, 6	mpg 1548	. . 3 |- (E!xE.y(x = A /\ ph) -> E!yE.x(x = A /\ ph))
8		2euswap 2280	. . . 4 |- (A.yE*x(x = A /\ ph) -> (E!yE.x(x = A /\ ph) -> E!xE.y(x = A /\ ph)))
9		moeq 3013	. . . . . 6 |- E*x x = A
10	9	moani 2256	. . . . 5 |- E*x(ph /\ x = A)
11	4	mobii 2240	. . . . 5 |- (E*x(ph /\ x = A) <-> E*x(x = A /\ ph))
12	10, 11	mpbi 199	. . . 4 |- E*x(x = A /\ ph)
13	8, 12	mpg 1548	. . 3 |- (E!yE.x(x = A /\ ph) -> E!xE.y(x = A /\ ph))
14	7, 13	impbii 180	. 2 |- (E!xE.y(x = A /\ ph) <-> E!yE.x(x = A /\ ph))
15		euxfr2.1	. . . 4 |- A e. _V
16		biidd 228	. . . 4 |- (x = A -> (ph <-> ph))
17	15, 16	ceqsexv 2895	. . 3 |- (E.x(x = A /\ ph) <-> ph)
18	17	eubii 2213	. 2 |- (E!yE.x(x = A /\ ph) <-> E!yph)
19	14, 18	bitri 240	1 |- (E!xE.y(x = A /\ ph) <-> E!yph)

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  E!weu 2204  E*wmo 2205  _Vcvv 2860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2862

This theorem is referenced by:  euxfr  3023