Theorem nfimdOLD 1809

Description: Obsolete proof of nfimd 1808 as of 29-Dec-2017. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nfimd.1	|- (ph -> F/xps)
nfimd.2	|- (ph -> F/xch)

Assertion

Ref	Expression
nfimdOLD	|- (ph -> F/x(ps -> ch))

Proof of Theorem nfimdOLD

Step	Hyp	Ref	Expression
1		nfimd.1	. 2 |- (ph -> F/xps)
2		nfimd.2	. 2 |- (ph -> F/xch)
3		nfa1 1788	. . . . 5 |- F/xA.x(ps -> A.xps)
4		hbnt 1775	. . . . . 6 |- (A.x(ps -> A.xps) -> (-. ps -> A.x -. ps))
5		pm2.21 100	. . . . . . . . . . 11 |- (-. ps -> (ps -> ch))
6	5	alimi 1559	. . . . . . . . . 10 |- (A.x -. ps -> A.x(ps -> ch))
7	6	imim2i 13	. . . . . . . . 9 |- ((-. ps -> A.x -. ps) -> (-. ps -> A.x(ps -> ch)))
8	7	adantr 451	. . . . . . . 8 |- (((-. ps -> A.x -. ps) /\ (ch -> A.xch)) -> (-. ps -> A.x(ps -> ch)))
9		ax-1 6	. . . . . . . . . . 11 |- (ch -> (ps -> ch))
10	9	alimi 1559	. . . . . . . . . 10 |- (A.xch -> A.x(ps -> ch))
11	10	imim2i 13	. . . . . . . . 9 |- ((ch -> A.xch) -> (ch -> A.x(ps -> ch)))
12	11	adantl 452	. . . . . . . 8 |- (((-. ps -> A.x -. ps) /\ (ch -> A.xch)) -> (ch -> A.x(ps -> ch)))
13	8, 12	jad 154	. . . . . . 7 |- (((-. ps -> A.x -. ps) /\ (ch -> A.xch)) -> ((ps -> ch) -> A.x(ps -> ch)))
14	13	ex 423	. . . . . 6 |- ((-. ps -> A.x -. ps) -> ((ch -> A.xch) -> ((ps -> ch) -> A.x(ps -> ch))))
15	4, 14	syl 15	. . . . 5 |- (A.x(ps -> A.xps) -> ((ch -> A.xch) -> ((ps -> ch) -> A.x(ps -> ch))))
16	3, 15	alimd 1764	. . . 4 |- (A.x(ps -> A.xps) -> (A.x(ch -> A.xch) -> A.x((ps -> ch) -> A.x(ps -> ch))))
17	16	imp 418	. . 3 |- ((A.x(ps -> A.xps) /\ A.x(ch -> A.xch)) -> A.x((ps -> ch) -> A.x(ps -> ch)))
18		df-nf 1545	. . . 4 |- ( F/xps <-> A.x(ps -> A.xps))
19		df-nf 1545	. . . 4 |- ( F/xch <-> A.x(ch -> A.xch))
20	18, 19	anbi12i 678	. . 3 |- (( F/xps /\ F/xch) <-> (A.x(ps -> A.xps) /\ A.x(ch -> A.xch)))
21		df-nf 1545	. . 3 |- ( F/x(ps -> ch) <-> A.x((ps -> ch) -> A.x(ps -> ch)))
22	17, 20, 21	3imtr4i 257	. 2 |- (( F/xps /\ F/xch) -> F/x(ps -> ch))
23	1, 2, 22	syl2anc 642	1 |- (ph -> F/x(ps -> ch))

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-11 1746

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

This theorem is referenced by: (None)