Theorem nfim1 2067

Description: A closed form of nfim 1825. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) df-nf 1710 changed. (Revised by Wolf Lammen, 18-Sep-2021.)

Hypotheses

Ref	Expression
nfim1.1	|- F/ x ph
nfim1.2	|- ( ph -> F/ x ps )

Assertion

Ref	Expression
nfim1	|- F/ x ( ph -> ps )

Proof of Theorem nfim1

Step	Hyp	Ref	Expression
1		nfim1.1	. . 3 |- F/ x ph
2		nf3 1712	. . 3 |- ( F/ x ph <-> ( A. x ph \/ A. x -. ph ) )
3	1, 2	mpbi 220	. 2 |- ( A. x ph \/ A. x -. ph )
4		nftht 1718	. . . 4 |- ( A. x ph -> F/ x ph )
5		nfim1.2	. . . . 5 |- ( ph -> F/ x ps )
6	5	sps 2055	. . . 4 |- ( A. x ph -> F/ x ps )
7	4, 6	nfimd 1823	. . 3 |- ( A. x ph -> F/ x ( ph -> ps ) )
8		pm2.21 120	. . . . 5 |- ( -. ph -> ( ph -> ps ) )
9	8	alimi 1739	. . . 4 |- ( A. x -. ph -> A. x ( ph -> ps ) )
10		nftht 1718	. . . 4 |- ( A. x ( ph -> ps ) -> F/ x ( ph -> ps ) )
11	9, 10	syl 17	. . 3 |- ( A. x -. ph -> F/ x ( ph -> ps ) )
12	7, 11	jaoi 394	. 2 |- ( ( A. x ph \/ A. x -. ph ) -> F/ x ( ph -> ps ) )
13	3, 12	ax-mp 5	1 |- F/ x ( ph -> ps )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   A.wal 1481   F/wnf 1708

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-12 2047

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710

This theorem is referenced by:  nfan1  2068  cbv1  2267  dvelimdf  2335  sbied  2409  sbco2d  2416  bj-cbv1v  32729