Theorem nfeudv 1915

Description: Deduction version of nfeu 1919. Similar to nfeud 1916 but has the additional constraint that x and y must be distinct. (Contributed by Jim Kingdon, 25-May-2018.)

Hypotheses

Ref	Expression
nfeudv.1	|- F/ y ph
nfeudv.2	|- ( ph -> F/ x ps )

Assertion

Ref	Expression
nfeudv	|- ( ph -> F/ x E! y ps )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)   ps( x, y)

Proof of Theorem nfeudv

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1421	. . 3 |- F/ z ph
2		nfeudv.1	. . . 4 |- F/ y ph
3		nfeudv.2	. . . . 5 |- ( ph -> F/ x ps )
4		nfv 1421	. . . . . 6 |- F/ x y = z
5	4	a1i 9	. . . . 5 |- ( ph -> F/ x y = z )
6	3, 5	nfbid 1480	. . . 4 |- ( ph -> F/ x ( ps <-> y = z ) )
7	2, 6	nfald 1643	. . 3 |- ( ph -> F/ x A. y ( ps <-> y = z ) )
8	1, 7	nfexd 1644	. 2 |- ( ph -> F/ x E. z A. y ( ps <-> y = z ) )
9		df-eu 1903	. . 3 |- ( E! y ps <-> E. z A. y ( ps <-> y = z ) )
10	9	nfbii 1362	. 2 |- ( F/ x E! y ps <-> F/ x E. z A. y ( ps <-> y = z ) )
11	8, 10	sylibr 137	1 |- ( ph -> F/ x E! y ps )

Syntax hints:   -> wi 4   <-> wb 98   A.wal 1241   = wceq 1243   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-nf 1350  df-eu 1903

This theorem is referenced by:  nfeud  1916