Theorem nfeud 1964

Description: Deduction version of nfeu 1967. (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 1466	. . 3 |- F/ z ps
2	1	sb8eu 1961	. 2 |- ( E! y ps <-> E! z [ z / y ] ps )
3		nfv 1466	. . 3 |- F/ z ph
4		nfeud.1	. . . 4 |- F/ y ph
5		nfeud.2	. . . 4 |- ( ph -> F/ x ps )
6	4, 5	nfsbd 1899	. . 3 |- ( ph -> F/ x [ z / y ] ps )
7	3, 6	nfeudv 1963	. 2 |- ( ph -> F/ x E! z [ z / y ] ps )
8	2, 7	nfxfrd 1409	1 |- ( ph -> F/ x E! y ps )

Syntax hints:   -> wi 4   F/wnf 1394   [wsb 1692   E!weu 1948

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951

This theorem is referenced by:  nfmod  1965  hbeud  1970  nfreudxy  2540