Theorem nfrexdxy 2399

Description: Not-free for restricted existential quantification where x and y are distinct. See nfrexdya 2401 for a version with y and A distinct instead. (Contributed by Jim Kingdon, 30-May-2018.)

Hypotheses

Ref	Expression
nfraldxy.2	|- F/ y ph
nfraldxy.3	|- ( ph -> F/_ x A )
nfraldxy.4	|- ( ph -> F/ x ps )

Assertion

Ref	Expression
nfrexdxy	|- ( ph -> F/ x E. y e. A ps )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)   ps( x, y)   A( x, y)

Proof of Theorem nfrexdxy

Step	Hyp	Ref	Expression
1		df-rex 2354	. 2 |- ( E. y e. A ps <-> E. y ( y e. A /\ ps ) )
2		nfraldxy.2	. . 3 |- F/ y ph
3		nfcv 2219	. . . . . 6 |- F/_ x y
4	3	a1i 9	. . . . 5 |- ( ph -> F/_ x y )
5		nfraldxy.3	. . . . 5 |- ( ph -> F/_ x A )
6	4, 5	nfeld 2234	. . . 4 |- ( ph -> F/ x y e. A )
7		nfraldxy.4	. . . 4 |- ( ph -> F/ x ps )
8	6, 7	nfand 1500	. . 3 |- ( ph -> F/ x ( y e. A /\ ps ) )
9	2, 8	nfexd 1684	. 2 |- ( ph -> F/ x E. y ( y e. A /\ ps ) )
10	1, 9	nfxfrd 1404	1 |- ( ph -> F/ x E. y e. A ps )

Syntax hints:   -> wi 4   /\ wa 102   F/wnf 1389   E.wex 1421   e. wcel 1433   F/_wnfc 2206   E.wrex 2349

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354

This theorem is referenced by:  nfrexdya  2401  nfrexxy  2403  nfunid  3608  strcollnft  10779