Theorem sbco3 2160

Description: A composition law for substitution. (Contributed by NM, 2-Jun-1993.) (Proof shortened by Wolf Lammen, 18-Sep-2018.)

Assertion

Ref	Expression
sbco3

Proof of Theorem sbco3

Step	Hyp	Ref	Expression
1		drsb1 2118	. . 3
2		nfae 2056	. . . 4
3		sbequ12a 1994	. . . . 5
4	3	sps 1865	. . . 4
5	2, 4	sbbid 2144	. . 3
6	1, 5	bitr3d 255	. 2
7		sbco 2154	. . . 4
8	7	sbbii 1746	. . 3
9		nfnae 2058	. . . 4
10		nfnae 2058	. . . 4
11		nfsb2 2100	. . . 4
12	9, 10, 11	sbco2d 2159	. . 3
13	8, 12	syl5rbbr 260	. 2
14	6, 13	pm2.61i 164	1

Syntax hints:  -.wn 3  <->wb 184  A.wal 1393  [wsb 1739

This theorem is referenced by:  sbcom  2161

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-or 370  df-an 371  df-ex 1613  df-nf 1617  df-sb 1740