Theorem nfeud 2040

Description: Deduction version of nfeu 2043. (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 1526	. . 3 |- F/ z ps
2	1	sb8eu 2037	. 2 |- ( E! y ps <-> E! z [ z / y ] ps )
3		nfv 1526	. . 3 |- F/ z ph
4		nfeud.1	. . . 4 |- F/ y ph
5		nfeud.2	. . . 4 |- ( ph -> F/ x ps )
6	4, 5	nfsbd 1975	. . 3 |- ( ph -> F/ x [ z / y ] ps )
7	3, 6	nfeudv 2039	. 2 |- ( ph -> F/ x E! z [ z / y ] ps )
8	2, 7	nfxfrd 1473	1 |- ( ph -> F/ x E! y ps )

Syntax hints:   -> wi 4   F/wnf 1458   [wsb 1760   E!weu 2024

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027

This theorem is referenced by:  nfmod  2041  hbeud  2046  nfreudxy  2648