Theorem nfeud2 2482

Description: Bound-variable hypothesis builder for uniqueness. (Contributed by Mario Carneiro, 14-Nov-2016.) (Proof shortened by Wolf Lammen, 4-Oct-2018.)

Hypotheses

Ref	Expression
nfeud2.1	|- F/ y ph
nfeud2.2	|- ( ( ph /\ -. A. x x = y ) -> F/ x ps )

Assertion

Ref	Expression
nfeud2	|- ( ph -> F/ x E! y ps )

Proof of Theorem nfeud2

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-eu 2474	. 2 |- ( E! y ps <-> E. z A. y ( ps <-> y = z ) )
2		nfv 1843	. . 3 |- F/ z ph
3		nfeud2.1	. . . 4 |- F/ y ph
4		nfeud2.2	. . . . 5 |- ( ( ph /\ -. A. x x = y ) -> F/ x ps )
5		nfeqf1 2299	. . . . . 6 |- ( -. A. x x = y -> F/ x y = z )
6	5	adantl 482	. . . . 5 |- ( ( ph /\ -. A. x x = y ) -> F/ x y = z )
7	4, 6	nfbid 1832	. . . 4 |- ( ( ph /\ -. A. x x = y ) -> F/ x ( ps <-> y = z ) )
8	3, 7	nfald2 2331	. . 3 |- ( ph -> F/ x A. y ( ps <-> y = z ) )
9	2, 8	nfexd 2167	. 2 |- ( ph -> F/ x E. z A. y ( ps <-> y = z ) )
10	1, 9	nfxfrd 1780	1 |- ( ph -> F/ x E! y ps )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   E.wex 1704   F/wnf 1708   E!weu 2470

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-eu 2474

This theorem is referenced by:  nfmod2  2483  nfeud  2484  nfreud  3112