Theorem nfald2 1972

Description: Variation on nfald 1852 which adds the hypothesis that x and y are distinct in the inner subproof. (Contributed by Mario Carneiro, 8-Oct-2016.)

Hypotheses

Ref	Expression
nfald2.1	|- F/yph
nfald2.2	|- ((ph /\ -. A.x x = y) -> F/xps)

Assertion

Ref	Expression
nfald2	|- (ph -> F/xA.yps)

Proof of Theorem nfald2

Step	Hyp	Ref	Expression
1		nfald2.1	. . . . 5 |- F/yph
2		nfnae 1956	. . . . 5 |- F/y -. A.x x = y
3	1, 2	nfan 1824	. . . 4 |- F/y(ph /\ -. A.x x = y)
4		nfald2.2	. . . 4 |- ((ph /\ -. A.x x = y) -> F/xps)
5	3, 4	nfald 1852	. . 3 |- ((ph /\ -. A.x x = y) -> F/xA.yps)
6	5	ex 423	. 2 |- (ph -> (-. A.x x = y -> F/xA.yps))
7		nfa1 1788	. . 3 |- F/yA.yps
8		biidd 228	. . . 4 |- (A.x x = y -> (A.yps <-> A.yps))
9	8	drnf1 1969	. . 3 |- (A.x x = y -> ( F/xA.yps <-> F/yA.yps))
10	7, 9	mpbiri 224	. 2 |- (A.x x = y -> F/xA.yps)
11	6, 10	pm2.61d2 152	1 |- (ph -> F/xA.yps)

Syntax hints:  -. wn 3   -> wi 4   /\ wa 358  A.wal 1540   F/wnf 1544

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

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by:  nfexd2  1973  dvelimf  1997  nfeud2  2216  nfrald  2666  nfiotad  4343