Theorem nfeud 1913

Description: Deduction version of nfeu 1916. (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/yph
nfeud.2	|- (ph -> F/xps)

Assertion

Ref	Expression
nfeud	|- (ph -> F/xE!yps)

Proof of Theorem nfeud

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1418	. . 3 |- F/zps
2	1	sb8eu 1910	. 2 |- (E!yps <-> E!z[z / y]ps)
3		nfv 1418	. . 3 |- F/zph
4		nfeud.1	. . . 4 |- F/yph
5		nfeud.2	. . . 4 |- (ph -> F/xps)
6	4, 5	nfsbd 1848	. . 3 |- (ph -> F/x[z / y]ps)
7	3, 6	nfeudv 1912	. 2 |- (ph -> F/xE!z[z / y]ps)
8	2, 7	nfxfrd 1361	1 |- (ph -> F/xE!yps)

Syntax hints:   -> wi 4   F/wnf 1346  [wsb 1642  E!weu 1897

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900

This theorem is referenced by:  nfmod  1914  hbeud  1919  nfreudxy  2477