Theorem sbal1 2204

Description: A theorem used in elimination of disjoint variable restriction on and by replacing it with a distinctor . (Contributed by NM, 15-May-1993.) (Proof shortened by Wolf Lammen, 3-Oct-2018.)

Assertion

Ref	Expression
sbal1

Distinct variable group:   ,

Proof of Theorem sbal1

Step	Hyp	Ref	Expression
1		sb4b 2098	. . . . 5
2		nfnae 2058	. . . . . 6
3		nfeqf2 2041	. . . . . . 7
4		19.21t 1904	. . . . . . . 8
5	4	bicomd 201	. . . . . . 7
6	3, 5	syl 16	. . . . . 6
7	2, 6	albid 1885	. . . . 5
8	1, 7	sylan9bbr 700	. . . 4
9		nfnae 2058	. . . . . . 7
10		sb4b 2098	. . . . . . 7
11	9, 10	albid 1885	. . . . . 6
12		alcom 1845	. . . . . 6
13	11, 12	syl6bb 261	. . . . 5
14	13	adantl 466	. . . 4
15	8, 14	bitr4d 256	. . 3
16	15	ex 434	. 2
17		sbequ12 1992	. . . 4
18	17	sps 1865	. . 3
19		sbequ12 1992	. . . . 5
20	19	sps 1865	. . . 4
21	20	dral2 2066	. . 3
22	18, 21	bitr3d 255	. 2
23	16, 22	pm2.61d2 160	1

Syntax hints:  -.wn 3  ->wi 4  <->wb 184  /\wa 369  A.wal 1393  F/wnf 1616  [wsb 1739

This theorem is referenced by:  sbal  2206

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