Theorem nfeud 2058

Description: Deduction version of nfeu 2061. (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 1539	. . 3 |- F/ z ps
2	1	sb8eu 2055	. 2 |- ( E! y ps <-> E! z [ z / y ] ps )
3		nfv 1539	. . 3 |- F/ z ph
4		nfeud.1	. . . 4 |- F/ y ph
5		nfeud.2	. . . 4 |- ( ph -> F/ x ps )
6	4, 5	nfsbd 1993	. . 3 |- ( ph -> F/ x [ z / y ] ps )
7	3, 6	nfeudv 2057	. 2 |- ( ph -> F/ x E! z [ z / y ] ps )
8	2, 7	nfxfrd 1486	1 |- ( ph -> F/ x E! y ps )

Syntax hints:   -> wi 4   F/wnf 1471   [wsb 1773   E!weu 2042

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045

This theorem is referenced by:  nfmod  2059  hbeud  2064  nfreudxy  2668