Theorem nfeuv 2032

Description: Bound-variable hypothesis builder for existential uniqueness. This is similar to nfeu 2033 but has the additional condition that x and y must be distinct. (Contributed by Jim Kingdon, 23-May-2018.)

Hypothesis

Ref	Expression
nfeuv.1	|- F/ x ph

Assertion

Ref	Expression
nfeuv	|- F/ x E! y ph

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem nfeuv

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfeuv.1	. . . . 5 |- F/ x ph
2		nfv 1516	. . . . 5 |- F/ x y = z
3	1, 2	nfbi 1577	. . . 4 |- F/ x ( ph <-> y = z )
4	3	nfal 1564	. . 3 |- F/ x A. y ( ph <-> y = z )
5	4	nfex 1625	. 2 |- F/ x E. z A. y ( ph <-> y = z )
6		df-eu 2017	. . 3 |- ( E! y ph <-> E. z A. y ( ph <-> y = z ) )
7	6	nfbii 1461	. 2 |- ( F/ x E! y ph <-> F/ x E. z A. y ( ph <-> y = z ) )
8	5, 7	mpbir 145	1 |- F/ x E! y ph

Syntax hints:   <-> wb 104   A.wal 1341   F/wnf 1448   E.wex 1480   E!weu 2014

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-17 1514  ax-ial 1522  ax-i5r 1523

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-eu 2017

This theorem is referenced by:  nfeu  2033