Theorem nfeuv 1918

Description: Bound-variable hypothesis builder for existential uniqueness. This is similar to nfeu 1919 but has the additional constraint 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 1421	. . . . 5 |- F/ x y = z
3	1, 2	nfbi 1481	. . . 4 |- F/ x ( ph <-> y = z )
4	3	nfal 1468	. . 3 |- F/ x A. y ( ph <-> y = z )
5	4	nfex 1528	. 2 |- F/ x E. z A. y ( ph <-> y = z )
6		df-eu 1903	. . 3 |- ( E! y ph <-> E. z A. y ( ph <-> y = z ) )
7	6	nfbii 1362	. 2 |- ( F/ x E! y ph <-> F/ x E. z A. y ( ph <-> y = z ) )
8	5, 7	mpbir 134	1 |- F/ x E! y ph

Syntax hints:   <-> wb 98   A.wal 1241   F/wnf 1349   E.wex 1381   E!weu 1900

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-eu 1903

This theorem is referenced by:  nfeu  1919