Theorem nfrexdxy 2405

Description: Not-free for restricted existential quantification where x and y are distinct. See nfrexdya 2407 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 2359	. 2 |- ( E. y e. A ps <-> E. y ( y e. A /\ ps ) )
2		nfraldxy.2	. . 3 |- F/ y ph
3		nfcv 2223	. . . . . 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 2238	. . . 4 |- ( ph -> F/ x y e. A )
7		nfraldxy.4	. . . 4 |- ( ph -> F/ x ps )
8	6, 7	nfand 1501	. . 3 |- ( ph -> F/ x ( y e. A /\ ps ) )
9	2, 8	nfexd 1686	. 2 |- ( ph -> F/ x E. y ( y e. A /\ ps ) )
10	1, 9	nfxfrd 1405	1 |- ( ph -> F/ x E. y e. A ps )

Syntax hints:   -> wi 4   /\ wa 102   F/wnf 1390   E.wex 1422   e. wcel 1434   F/_wnfc 2210   E.wrex 2354

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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-nf 1391  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rex 2359

This theorem is referenced by:  nfrexdya  2407  nfrexxy  2409  nfunid  3634  strcollnft  11220