Theorem nfeudv 2014

Description: Deduction version of nfeu 2018. Similar to nfeud 2015 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 1508	. . 3 |- F/ z ph
2		nfeudv.1	. . . 4 |- F/ y ph
3		nfeudv.2	. . . . 5 |- ( ph -> F/ x ps )
4		nfv 1508	. . . . . 6 |- F/ x y = z
5	4	a1i 9	. . . . 5 |- ( ph -> F/ x y = z )
6	3, 5	nfbid 1567	. . . 4 |- ( ph -> F/ x ( ps <-> y = z ) )
7	2, 6	nfald 1733	. . 3 |- ( ph -> F/ x A. y ( ps <-> y = z ) )
8	1, 7	nfexd 1734	. 2 |- ( ph -> F/ x E. z A. y ( ps <-> y = z ) )
9		df-eu 2002	. . 3 |- ( E! y ps <-> E. z A. y ( ps <-> y = z ) )
10	9	nfbii 1449	. 2 |- ( F/ x E! y ps <-> F/ x E. z A. y ( ps <-> y = z ) )
11	8, 10	sylibr 133	1 |- ( ph -> F/ x E! y ps )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1329   = wceq 1331   F/wnf 1436   E.wex 1468   E!weu 1999

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-i5r 1515

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-eu 2002

This theorem is referenced by:  nfeud  2015