Theorem nfraldxy 2470

Description: Not-free for restricted universal quantification where x and y are distinct. See nfraldya 2472 for a version with y and A distinct instead. (Contributed by Jim Kingdon, 29-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
nfraldxy	|- ( ph -> F/ x A. y e. A ps )

Distinct variable group:   x, y

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

Proof of Theorem nfraldxy

Step	Hyp	Ref	Expression
1		df-ral 2422	. 2 |- ( A. y e. A ps <-> A. y ( y e. A -> ps ) )
2		nfraldxy.2	. . 3 |- F/ y ph
3		nfcv 2282	. . . . . 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 2298	. . . 4 |- ( ph -> F/ x y e. A )
7		nfraldxy.4	. . . 4 |- ( ph -> F/ x ps )
8	6, 7	nfimd 1565	. . 3 |- ( ph -> F/ x ( y e. A -> ps ) )
9	2, 8	nfald 1734	. 2 |- ( ph -> F/ x A. y ( y e. A -> ps ) )
10	1, 9	nfxfrd 1452	1 |- ( ph -> F/ x A. y e. A ps )

Syntax hints:   -> wi 4   A.wal 1330   F/wnf 1437   e. wcel 1481   F/_wnfc 2269   A.wral 2417

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-17 1507  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422

This theorem is referenced by:  nfraldya  2472  nfralxy  2474  strcollnft  13353