Theorem nfeud 1916

Description: Deduction version of nfeu 1919. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof rewritten by Jim Kingdon, 25-May-2018.)

Hypotheses

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

Assertion

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

Proof of Theorem nfeud

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1421	. . 3 |- F/ z ps
2	1	sb8eu 1913	. 2 |- ( E! y ps <-> E! z [ z / y ] ps )
3		nfv 1421	. . 3 |- F/ z ph
4		nfeud.1	. . . 4 |- F/ y ph
5		nfeud.2	. . . 4 |- ( ph -> F/ x ps )
6	4, 5	nfsbd 1851	. . 3 |- ( ph -> F/ x [ z / y ] ps )
7	3, 6	nfeudv 1915	. 2 |- ( ph -> F/ x E! z [ z / y ] ps )
8	2, 7	nfxfrd 1364	1 |- ( ph -> F/ x E! y ps )

Syntax hints:   -> wi 4   F/wnf 1349   [wsb 1645   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-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903

This theorem is referenced by:  nfmod  1917  hbeud  1922  nfreudxy  2483