Theorem nfexd2 2332

Description: Variation on nfexd 2167 which adds the hypothesis that x and y are distinct in the inner subproof. (Contributed by Mario Carneiro, 8-Oct-2016.)

Hypotheses

Ref	Expression
nfald2.1	|- F/ y ph
nfald2.2	|- ( ( ph /\ -. A. x x = y ) -> F/ x ps )

Assertion

Ref	Expression
nfexd2	|- ( ph -> F/ x E. y ps )

Proof of Theorem nfexd2

Step	Hyp	Ref	Expression
1		df-ex 1705	. 2 |- ( E. y ps <-> -. A. y -. ps )
2		nfald2.1	. . . 4 |- F/ y ph
3		nfald2.2	. . . . 5 |- ( ( ph /\ -. A. x x = y ) -> F/ x ps )
4	3	nfnd 1785	. . . 4 |- ( ( ph /\ -. A. x x = y ) -> F/ x -. ps )
5	2, 4	nfald2 2331	. . 3 |- ( ph -> F/ x A. y -. ps )
6	5	nfnd 1785	. 2 |- ( ph -> F/ x -. A. y -. ps )
7	1, 6	nfxfrd 1780	1 |- ( ph -> F/ x E. y ps )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   A.wal 1481   E.wex 1704   F/wnf 1708

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

This theorem is referenced by:  nfmod2  2483