Theorem somincom 5530

Description: Commutativity of minimum in a total order. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Assertion

Ref	Expression
somincom	|- ( ( R Or X /\ ( A e. X /\ B e. X ) ) -> if ( A R B , A , B ) = if ( B R A , B , A ) )

Proof of Theorem somincom

Step	Hyp	Ref	Expression
1		so2nr 5059	. . . . 5 |- ( ( R Or X /\ ( A e. X /\ B e. X ) ) -> -. ( A R B /\ B R A ) )
2		nan 604	. . . . 5 |- ( ( ( R Or X /\ ( A e. X /\ B e. X ) ) -> -. ( A R B /\ B R A ) ) <-> ( ( ( R Or X /\ ( A e. X /\ B e. X ) ) /\ A R B ) -> -. B R A ) )
3	1, 2	mpbi 220	. . . 4 |- ( ( ( R Or X /\ ( A e. X /\ B e. X ) ) /\ A R B ) -> -. B R A )
4	3	iffalsed 4097	. . 3 |- ( ( ( R Or X /\ ( A e. X /\ B e. X ) ) /\ A R B ) -> if ( B R A , B , A ) = A )
5	4	eqcomd 2628	. 2 |- ( ( ( R Or X /\ ( A e. X /\ B e. X ) ) /\ A R B ) -> A = if ( B R A , B , A ) )
6		sotric 5061	. . . . 5 |- ( ( R Or X /\ ( A e. X /\ B e. X ) ) -> ( A R B <-> -. ( A = B \/ B R A ) ) )
7	6	con2bid 344	. . . 4 |- ( ( R Or X /\ ( A e. X /\ B e. X ) ) -> ( ( A = B \/ B R A ) <-> -. A R B ) )
8		ifeq2 4091	. . . . . 6 |- ( A = B -> if ( B R A , B , A ) = if ( B R A , B , B ) )
9		ifid 4125	. . . . . 6 |- if ( B R A , B , B ) = B
10	8, 9	syl6req 2673	. . . . 5 |- ( A = B -> B = if ( B R A , B , A ) )
11		iftrue 4092	. . . . . 6 |- ( B R A -> if ( B R A , B , A ) = B )
12	11	eqcomd 2628	. . . . 5 |- ( B R A -> B = if ( B R A , B , A ) )
13	10, 12	jaoi 394	. . . 4 |- ( ( A = B \/ B R A ) -> B = if ( B R A , B , A ) )
14	7, 13	syl6bir 244	. . 3 |- ( ( R Or X /\ ( A e. X /\ B e. X ) ) -> ( -. A R B -> B = if ( B R A , B , A ) ) )
15	14	imp 445	. 2 |- ( ( ( R Or X /\ ( A e. X /\ B e. X ) ) /\ -. A R B ) -> B = if ( B R A , B , A ) )
16	5, 15	ifeqda 4121	1 |- ( ( R Or X /\ ( A e. X /\ B e. X ) ) -> if ( A R B , A , B ) = if ( B R A , B , A ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   ifcif 4086   class class class wbr 4653   Or wor 5034

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-po 5035  df-so 5036

This theorem is referenced by:  somin2  5531