Theorem nfeuv 1959

Description: Bound-variable hypothesis builder for existential uniqueness. This is similar to nfeu 1960 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 1461	. . . . 5 |- F/ x y = z
3	1, 2	nfbi 1521	. . . 4 |- F/ x ( ph <-> y = z )
4	3	nfal 1508	. . 3 |- F/ x A. y ( ph <-> y = z )
5	4	nfex 1568	. 2 |- F/ x E. z A. y ( ph <-> y = z )
6		df-eu 1944	. . 3 |- ( E! y ph <-> E. z A. y ( ph <-> y = z ) )
7	6	nfbii 1402	. 2 |- ( F/ x E! y ph <-> F/ x E. z A. y ( ph <-> y = z ) )
8	5, 7	mpbir 144	1 |- F/ x E! y ph

Syntax hints:   <-> wb 103   A.wal 1282   F/wnf 1389   E.wex 1421   E!weu 1941

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-eu 1944

This theorem is referenced by:  nfeu  1960