Theorem sbnf2 2183

Description: Two ways of expressing " is (effectively) not free in ." (Contributed by Gérard Lang, 14-Nov-2013.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 22-Sep-2018.)

Assertion

Ref	Expression
sbnf2

Distinct variable groups:   ,,   ,,

Proof of Theorem sbnf2

Step	Hyp	Ref	Expression
1		nfv 1707	. . . . . 6
2	1	sb8e 2168	. . . . 5
3		nfv 1707	. . . . . 6
4	3	sb8 2167	. . . . 5
5	2, 4	imbi12i 326	. . . 4
6		nf2 1960	. . . 4
7		pm11.53v 1764	. . . 4
8	5, 6, 7	3bitr4i 277	. . 3
9	3	sb8e 2168	. . . . . 6
10	1	sb8 2167	. . . . . 6
11	9, 10	imbi12i 326	. . . . 5
12		pm11.53v 1764	. . . . 5
13	11, 12	bitr4i 252	. . . 4
14		alcom 1845	. . . 4
15	6, 13, 14	3bitri 271	. . 3
16	8, 15	anbi12i 697	. 2
17		pm4.24 643	. 2
18		2albiim 1700	. 2
19	16, 17, 18	3bitr4i 277	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  A.wal 1393  E.wex 1612  F/wnf 1616  [wsb 1739

This theorem is referenced by:  sbnfc2  3854  nfnid  4681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999

This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1613  df-nf 1617  df-sb 1740