Theorem sbcnestgf 3839

Description: Nest the composition of two substitutions. (Contributed by Mario Carneiro, 11-Nov-2016.)

Assertion

Ref	Expression
sbcnestgf

Proof of Theorem sbcnestgf

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq 3329	. . . . 5
2		csbeq1 3437	. . . . . 6
3	2	sbceq1d 3332	. . . . 5
4	1, 3	bibi12d 321	. . . 4
5	4	imbi2d 316	. . 3
6		vex 3112	. . . . 5
7	6	a1i 11	. . . 4
8		csbeq1a 3443	. . . . . 6
9	8	sbceq1d 3332	. . . . 5
10	9	adantl 466	. . . 4
11		nfnf1 1899	. . . . 5
12	11	nfal 1947	. . . 4
13		nfa1 1897	. . . . 5
14		nfcsb1v 3450	. . . . . 6
15	14	a1i 11	. . . . 5
16		sp 1859	. . . . 5
17	13, 15, 16	nfsbcd 3348	. . . 4
18	7, 10, 12, 17	sbciedf 3363	. . 3
19	5, 18	vtoclg 3167	. 2
20	19	imp 429	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  A.wal 1393  =wceq 1395  F/wnf 1616  e.wcel 1818  F/_wnfc 2605  cvv 3109  [.wsbc 3327  [_csb 3434

This theorem is referenced by:  csbnestgf  3840  sbcnestg  3841

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  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-sbc 3328  df-csb 3435