Theorem sbceqg 3825

Description: Distribute proper substitution through an equality relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Assertion

Ref	Expression
sbceqg

Proof of Theorem sbceqg

Dummy variables are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq2 3330	. . 3
2		dfsbcq2 3330	. . . . 5
3	2	abbidv 2593	. . . 4
4		dfsbcq2 3330	. . . . 5
5	4	abbidv 2593	. . . 4
6	3, 5	eqeq12d 2479	. . 3
7		nfs1v 2181	. . . . . 6
8	7	nfab 2623	. . . . 5
9		nfs1v 2181	. . . . . 6
10	9	nfab 2623	. . . . 5
11	8, 10	nfeq 2630	. . . 4
12		sbab 2604	. . . . 5
13		sbab 2604	. . . . 5
14	12, 13	eqeq12d 2479	. . . 4
15	11, 14	sbie 2149	. . 3
16	1, 6, 15	vtoclbg 3168	. 2
17		df-csb 3435	. . 3
18		df-csb 3435	. . 3
19	17, 18	eqeq12i 2477	. 2
20	16, 19	syl6bbr 263	1

Syntax hints:  ->wi 4  <->wb 184  =wceq 1395  [wsb 1739  e.wcel 1818  {cab 2442  [.wsbc 3327  [_csb 3434

This theorem is referenced by:  sbcne12  3827  sbcne12gOLD  3828  sbceq1g  3830  sbceq2g  3833  sbcfng  5733  swrdspsleq  12673  sbceqi  30513  onfrALTlem5  33314  onfrALTlem4  33315  csbeq2gOLD  33322  csbfv12gALTOLD  33621  csbingVD  33684  onfrALTlem5VD  33685  onfrALTlem4VD  33686  csbeq2gVD  33692  csbsngVD  33693  csbunigVD  33698  csbfv12gALTVD  33699  cdlemk42  36667

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-or 370  df-an 371  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