Theorem nfrald 2666

Description: Deduction version of nfral 2668. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 7-Oct-2016.)

Hypotheses

Ref	Expression
nfrald.2	|- F/yph
nfrald.3	|- (ph -> F/_xA)
nfrald.4	|- (ph -> F/xps)

Assertion

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

Proof of Theorem nfrald

Step	Hyp	Ref	Expression
1		df-ral 2620	. 2 |- (A.y e. A ps <-> A.y(y e. A -> ps))
2		nfrald.2	. . 3 |- F/yph
3		nfcvf 2512	. . . . . 6 |- (-. A.x x = y -> F/_xy)
4	3	adantl 452	. . . . 5 |- ((ph /\ -. A.x x = y) -> F/_xy)
5		nfrald.3	. . . . . 6 |- (ph -> F/_xA)
6	5	adantr 451	. . . . 5 |- ((ph /\ -. A.x x = y) -> F/_xA)
7	4, 6	nfeld 2505	. . . 4 |- ((ph /\ -. A.x x = y) -> F/x y e. A)
8		nfrald.4	. . . . 5 |- (ph -> F/xps)
9	8	adantr 451	. . . 4 |- ((ph /\ -. A.x x = y) -> F/xps)
10	7, 9	nfimd 1808	. . 3 |- ((ph /\ -. A.x x = y) -> F/x(y e. A -> ps))
11	2, 10	nfald2 1972	. 2 |- (ph -> F/xA.y(y e. A -> ps))
12	1, 11	nfxfrd 1571	1 |- (ph -> F/xA.y e. A ps)

Syntax hints:  -. wn 3   -> wi 4   /\ wa 358  A.wal 1540   F/wnf 1544   = wceq 1642   e. wcel 1710   F/_wnfc 2477  A.wral 2615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620

This theorem is referenced by:  nfrexd  2667  nfral  2668