Theorem nfeuv 1960

Description: Bound-variable hypothesis builder for existential uniqueness. This is similar to nfeu 1961 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 1462	. . . . 5 |- F/ x y = z
3	1, 2	nfbi 1522	. . . 4 |- F/ x ( ph <-> y = z )
4	3	nfal 1509	. . 3 |- F/ x A. y ( ph <-> y = z )
5	4	nfex 1569	. 2 |- F/ x E. z A. y ( ph <-> y = z )
6		df-eu 1945	. . 3 |- ( E! y ph <-> E. z A. y ( ph <-> y = z ) )
7	6	nfbii 1403	. 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 1283   F/wnf 1390   E.wex 1422   E!weu 1942

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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468  ax-i5r 1469

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-eu 1945

This theorem is referenced by:  nfeu  1961