Theorem nfraldxy 2530

Description: Old name for nfraldw 2529. (Contributed by Jim Kingdon, 29-May-2018.) (New usage is discouraged.)

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 2480	. 2 |- ( A. y e. A ps <-> A. y ( y e. A -> ps ) )
2		nfraldxy.2	. . 3 |- F/ y ph
3		nfcv 2339	. . . . . 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 2355	. . . 4 |- ( ph -> F/ x y e. A )
7		nfraldxy.4	. . . 4 |- ( ph -> F/ x ps )
8	6, 7	nfimd 1599	. . 3 |- ( ph -> F/ x ( y e. A -> ps ) )
9	2, 8	nfald 1774	. 2 |- ( ph -> F/ x A. y ( y e. A -> ps ) )
10	1, 9	nfxfrd 1489	1 |- ( ph -> F/ x A. y e. A ps )

Syntax hints:   -> wi 4   A.wal 1362   F/wnf 1474   e. wcel 2167   F/_wnfc 2326   A.wral 2475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480

This theorem is referenced by:  nfraldya  2532  nfralxy  2535  strcollnft  15630