Theorem euxfr 3392

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! x ph <-> E! y ps )

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 2494	. . . . . 6 |- ( E! y x = A -> E. y x = A )
3	1, 2	ax-mp 5	. . . . 5 |- E. y x = A
4	3	biantrur 527	. . . 4 |- ( ph <-> ( E. y x = A /\ ph ) )
5		19.41v 1914	. . . 4 |- ( E. y ( x = A /\ ph ) <-> ( E. y x = A /\ ph ) )
6		euxfr.3	. . . . . 6 |- ( x = A -> ( ph <-> ps ) )
7	6	pm5.32i 669	. . . . 5 |- ( ( x = A /\ ph ) <-> ( x = A /\ ps ) )
8	7	exbii 1774	. . . 4 |- ( E. y ( x = A /\ ph ) <-> E. y ( x = A /\ ps ) )
9	4, 5, 8	3bitr2i 288	. . 3 |- ( ph <-> E. y ( x = A /\ ps ) )
10	9	eubii 2492	. 2 |- ( E! x ph <-> E! x E. y ( x = A /\ ps ) )
11		euxfr.1	. . 3 |- A e. _V
12	1	eumoi 2500	. . 3 |- E* y x = A
13	11, 12	euxfr2 3391	. 2 |- ( E! x E. y ( x = A /\ ps ) <-> E! y ps )
14	10, 13	bitri 264	1 |- ( E! x ph <-> E! y ps )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   E!weu 2470   _Vcvv 3200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202

This theorem is referenced by:  moxfr  37255