Theorem nfrald 2944

Description: Deduction version of nfral 2945. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 7-Oct-2016.)

Hypotheses

Ref	Expression
nfrald.1	|- F/ y ph
nfrald.2	|- ( ph -> F/_ x A )
nfrald.3	|- ( ph -> F/ x ps )

Assertion

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

Proof of Theorem nfrald

Step	Hyp	Ref	Expression
1		df-ral 2917	. 2 |- ( A. y e. A ps <-> A. y ( y e. A -> ps ) )
2		nfrald.1	. . 3 |- F/ y ph
3		nfcvf 2788	. . . . . 6 |- ( -. A. x x = y -> F/_ x y )
4	3	adantl 482	. . . . 5 |- ( ( ph /\ -. A. x x = y ) -> F/_ x y )
5		nfrald.2	. . . . . 6 |- ( ph -> F/_ x A )
6	5	adantr 481	. . . . 5 |- ( ( ph /\ -. A. x x = y ) -> F/_ x A )
7	4, 6	nfeld 2773	. . . 4 |- ( ( ph /\ -. A. x x = y ) -> F/ x y e. A )
8		nfrald.3	. . . . 5 |- ( ph -> F/ x ps )
9	8	adantr 481	. . . 4 |- ( ( ph /\ -. A. x x = y ) -> F/ x ps )
10	7, 9	nfimd 1823	. . 3 |- ( ( ph /\ -. A. x x = y ) -> F/ x ( y e. A -> ps ) )
11	2, 10	nfald2 2331	. 2 |- ( ph -> F/ x A. y ( y e. A -> ps ) )
12	1, 11	nfxfrd 1780	1 |- ( ph -> F/ x A. y e. A ps )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   A.wal 1481   F/wnf 1708   e. wcel 1990   F/_wnfc 2751   A.wral 2912

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917

This theorem is referenced by:  nfral  2945  nfrexd  3006