Theorem nfeuv 2044

Description: Bound-variable hypothesis builder for existential uniqueness. This is similar to nfeu 2045 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 1528	. . . . 5 |- F/ x y = z
3	1, 2	nfbi 1589	. . . 4 |- F/ x ( ph <-> y = z )
4	3	nfal 1576	. . 3 |- F/ x A. y ( ph <-> y = z )
5	4	nfex 1637	. 2 |- F/ x E. z A. y ( ph <-> y = z )
6		df-eu 2029	. . 3 |- ( E! y ph <-> E. z A. y ( ph <-> y = z ) )
7	6	nfbii 1473	. 2 |- ( F/ x E! y ph <-> F/ x E. z A. y ( ph <-> y = z ) )
8	5, 7	mpbir 146	1 |- F/ x E! y ph

Syntax hints:   <-> wb 105   A.wal 1351   F/wnf 1460   E.wex 1492   E!weu 2026

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534  ax-i5r 1535

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-eu 2029

This theorem is referenced by:  nfeu  2045