Theorem nfeud 2016

Description: Deduction version of nfeu 2019. (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 1509	. . 3 |- F/ z ps
2	1	sb8eu 2013	. 2 |- ( E! y ps <-> E! z [ z / y ] ps )
3		nfv 1509	. . 3 |- F/ z ph
4		nfeud.1	. . . 4 |- F/ y ph
5		nfeud.2	. . . 4 |- ( ph -> F/ x ps )
6	4, 5	nfsbd 1951	. . 3 |- ( ph -> F/ x [ z / y ] ps )
7	3, 6	nfeudv 2015	. 2 |- ( ph -> F/ x E! z [ z / y ] ps )
8	2, 7	nfxfrd 1452	1 |- ( ph -> F/ x E! y ps )

Syntax hints:   -> wi 4   F/wnf 1437   [wsb 1736   E!weu 2000

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-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003

This theorem is referenced by:  nfmod  2017  hbeud  2022  nfreudxy  2607