Theorem sbccom 3407

Description: Commutative law for double class substitution. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Mario Carneiro, 18-Oct-2016.)

Assertion

Ref	Expression
sbccom

Distinct variable groups:   ,   ,   ,

Proof of Theorem sbccom

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

Step	Hyp	Ref	Expression
1		sbccomlem 3406	. . . 4
2		sbccomlem 3406	. . . . . . 7
3	2	sbcbii 3387	. . . . . 6
4		sbccomlem 3406	. . . . . 6
5	3, 4	bitri 249	. . . . 5
6	5	sbcbii 3387	. . . 4
7		sbccomlem 3406	. . . . 5
8	7	sbcbii 3387	. . . 4
9	1, 6, 8	3bitr3i 275	. . 3
10		sbcco 3350	. . 3
11		sbcco 3350	. . 3
12	9, 10, 11	3bitr3i 275	. 2
13		sbcco 3350	. . 3
14	13	sbcbii 3387	. 2
15		sbcco 3350	. . 3
16	15	sbcbii 3387	. 2
17	12, 14, 16	3bitr3i 275	1

Syntax hints:  <->wb 184  [.wsbc 3327

This theorem is referenced by:  csbcom  3837  csbcomgOLD  3838  csbab  3855  csbabgOLD  3856  mpt2xopovel  6967  wrd2ind  12703  elmptrab  20328  sbccom2  30530  sbcrot3  30724

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-v 3111  df-sbc 3328