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

Assertion

Ref	Expression
euxfr	|- (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 euxfr

Step	Hyp	Ref	Expression
1		euxfr.2	. . . . . 6 |- E!y x = A
2		euex 2227	. . . . . 6 |- (E!y x = A -> E.y x = A)
3	1, 2	ax-mp 5	. . . . 5 |- E.y x = A
4	3	biantrur 492	. . . 4 |- (ph <-> (E.y x = A /\ ph))
5		19.41v 1901	. . . 4 |- (E.y(x = A /\ ph) <-> (E.y x = A /\ ph))
6		euxfr.3	. . . . . 6 |- (x = A -> (ph <-> ps))
7	6	pm5.32i 618	. . . . 5 |- ((x = A /\ ph) <-> (x = A /\ ps))
8	7	exbii 1582	. . . 4 |- (E.y(x = A /\ ph) <-> E.y(x = A /\ ps))
9	4, 5, 8	3bitr2i 264	. . 3 |- (ph <-> E.y(x = A /\ ps))
10	9	eubii 2213	. 2 |- (E!xph <-> E!xE.y(x = A /\ ps))
11		euxfr.1	. . 3 |- A e. _V
12	1	eumoi 2245	. . 3 |- E*y x = A
13	11, 12	euxfr2 3021	. 2 |- (E!xE.y(x = A /\ ps) <-> E!yps)
14	10, 13	bitri 240	1 |- (E!xph <-> E!yps)

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  E!weu 2204  _Vcvv 2859

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 2861

This theorem is referenced by: (None)