Theorem setscomd 13068

Description: Different components can be set in any order. (Contributed by Jim Kingdon, 20-Feb-2025.)

Hypotheses

Ref	Expression
setscomd.a	|- ( ph -> A e. Y )
setscomd.b	|- ( ph -> B e. Z )
setscomd.s	|- ( ph -> S e. V )
setscomd.ab	|- ( ph -> A =/= B )
setscomd.c	|- ( ph -> C e. W )
setscomd.d	|- ( ph -> D e. X )

Assertion

Ref	Expression
setscomd	|- ( ph -> ( ( S sSet <. A , C >. ) sSet <. B , D >. ) = ( ( S sSet <. B , D >. ) sSet <. A , C >. ) )

Proof of Theorem setscomd

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

Step	Hyp	Ref	Expression
1		setscomd.ab	. 2 |- ( ph -> A =/= B )
2		setscomd.b	. . 3 |- ( ph -> B e. Z )
3		simpr 110	. . . . 5 |- ( ( ph /\ b = B ) -> b = B )
4	3	neeq2d 2419	. . . 4 |- ( ( ph /\ b = B ) -> ( A =/= b <-> A =/= B ) )
5	3	opeq1d 3862	. . . . . 6 |- ( ( ph /\ b = B ) -> <. b , D >. = <. B , D >. )
6	5	oveq2d 6016	. . . . 5 |- ( ( ph /\ b = B ) -> ( ( S sSet <. A , C >. ) sSet <. b , D >. ) = ( ( S sSet <. A , C >. ) sSet <. B , D >. ) )
7	5	oveq2d 6016	. . . . . 6 |- ( ( ph /\ b = B ) -> ( S sSet <. b , D >. ) = ( S sSet <. B , D >. ) )
8	7	oveq1d 6015	. . . . 5 |- ( ( ph /\ b = B ) -> ( ( S sSet <. b , D >. ) sSet <. A , C >. ) = ( ( S sSet <. B , D >. ) sSet <. A , C >. ) )
9	6, 8	eqeq12d 2244	. . . 4 |- ( ( ph /\ b = B ) -> ( ( ( S sSet <. A , C >. ) sSet <. b , D >. ) = ( ( S sSet <. b , D >. ) sSet <. A , C >. ) <-> ( ( S sSet <. A , C >. ) sSet <. B , D >. ) = ( ( S sSet <. B , D >. ) sSet <. A , C >. ) ) )
10	4, 9	imbi12d 234	. . 3 |- ( ( ph /\ b = B ) -> ( ( A =/= b -> ( ( S sSet <. A , C >. ) sSet <. b , D >. ) = ( ( S sSet <. b , D >. ) sSet <. A , C >. ) ) <-> ( A =/= B -> ( ( S sSet <. A , C >. ) sSet <. B , D >. ) = ( ( S sSet <. B , D >. ) sSet <. A , C >. ) ) ) )
11		setscomd.a	. . . 4 |- ( ph -> A e. Y )
12		simpr 110	. . . . . 6 |- ( ( ph /\ a = A ) -> a = A )
13	12	neeq1d 2418	. . . . 5 |- ( ( ph /\ a = A ) -> ( a =/= b <-> A =/= b ) )
14	12	opeq1d 3862	. . . . . . . 8 |- ( ( ph /\ a = A ) -> <. a , C >. = <. A , C >. )
15	14	oveq2d 6016	. . . . . . 7 |- ( ( ph /\ a = A ) -> ( S sSet <. a , C >. ) = ( S sSet <. A , C >. ) )
16	15	oveq1d 6015	. . . . . 6 |- ( ( ph /\ a = A ) -> ( ( S sSet <. a , C >. ) sSet <. b , D >. ) = ( ( S sSet <. A , C >. ) sSet <. b , D >. ) )
17	14	oveq2d 6016	. . . . . 6 |- ( ( ph /\ a = A ) -> ( ( S sSet <. b , D >. ) sSet <. a , C >. ) = ( ( S sSet <. b , D >. ) sSet <. A , C >. ) )
18	16, 17	eqeq12d 2244	. . . . 5 |- ( ( ph /\ a = A ) -> ( ( ( S sSet <. a , C >. ) sSet <. b , D >. ) = ( ( S sSet <. b , D >. ) sSet <. a , C >. ) <-> ( ( S sSet <. A , C >. ) sSet <. b , D >. ) = ( ( S sSet <. b , D >. ) sSet <. A , C >. ) ) )
19	13, 18	imbi12d 234	. . . 4 |- ( ( ph /\ a = A ) -> ( ( a =/= b -> ( ( S sSet <. a , C >. ) sSet <. b , D >. ) = ( ( S sSet <. b , D >. ) sSet <. a , C >. ) ) <-> ( A =/= b -> ( ( S sSet <. A , C >. ) sSet <. b , D >. ) = ( ( S sSet <. b , D >. ) sSet <. A , C >. ) ) ) )
20		setscomd.s	. . . . . . 7 |- ( ph -> S e. V )
21	20	adantr 276	. . . . . 6 |- ( ( ph /\ a =/= b ) -> S e. V )
22		simpr 110	. . . . . 6 |- ( ( ph /\ a =/= b ) -> a =/= b )
23		setscomd.c	. . . . . . 7 |- ( ph -> C e. W )
24	23	adantr 276	. . . . . 6 |- ( ( ph /\ a =/= b ) -> C e. W )
25		setscomd.d	. . . . . . 7 |- ( ph -> D e. X )
26	25	adantr 276	. . . . . 6 |- ( ( ph /\ a =/= b ) -> D e. X )
27		vex 2802	. . . . . . 7 |- a e. _V
28		vex 2802	. . . . . . 7 |- b e. _V
29	27, 28	setscom 13067	. . . . . 6 |- ( ( ( S e. V /\ a =/= b ) /\ ( C e. W /\ D e. X ) ) -> ( ( S sSet <. a , C >. ) sSet <. b , D >. ) = ( ( S sSet <. b , D >. ) sSet <. a , C >. ) )
30	21, 22, 24, 26, 29	syl22anc 1272	. . . . 5 |- ( ( ph /\ a =/= b ) -> ( ( S sSet <. a , C >. ) sSet <. b , D >. ) = ( ( S sSet <. b , D >. ) sSet <. a , C >. ) )
31	30	ex 115	. . . 4 |- ( ph -> ( a =/= b -> ( ( S sSet <. a , C >. ) sSet <. b , D >. ) = ( ( S sSet <. b , D >. ) sSet <. a , C >. ) ) )
32	11, 19, 31	vtocld 2853	. . 3 |- ( ph -> ( A =/= b -> ( ( S sSet <. A , C >. ) sSet <. b , D >. ) = ( ( S sSet <. b , D >. ) sSet <. A , C >. ) ) )
33	2, 10, 32	vtocld 2853	. 2 |- ( ph -> ( A =/= B -> ( ( S sSet <. A , C >. ) sSet <. B , D >. ) = ( ( S sSet <. B , D >. ) sSet <. A , C >. ) ) )
34	1, 33	mpd 13	1 |- ( ph -> ( ( S sSet <. A , C >. ) sSet <. B , D >. ) = ( ( S sSet <. B , D >. ) sSet <. A , C >. ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   =/= wne 2400   <.cop 3669  (class class class)co 6000   sSet csts 13025

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-res 4730  df-iota 5277  df-fun 5319  df-fv 5325  df-ov 6003  df-oprab 6004  df-mpo 6005  df-sets 13034

This theorem is referenced by:  mgpress  13889