Theorem setscomd 12659

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 2383	. . . 4 |- ( ( ph /\ b = B ) -> ( A =/= b <-> A =/= B ) )
5	3	opeq1d 3810	. . . . . 6 |- ( ( ph /\ b = B ) -> <. b , D >. = <. B , D >. )
6	5	oveq2d 5934	. . . . 5 |- ( ( ph /\ b = B ) -> ( ( S sSet <. A , C >. ) sSet <. b , D >. ) = ( ( S sSet <. A , C >. ) sSet <. B , D >. ) )
7	5	oveq2d 5934	. . . . . 6 |- ( ( ph /\ b = B ) -> ( S sSet <. b , D >. ) = ( S sSet <. B , D >. ) )
8	7	oveq1d 5933	. . . . 5 |- ( ( ph /\ b = B ) -> ( ( S sSet <. b , D >. ) sSet <. A , C >. ) = ( ( S sSet <. B , D >. ) sSet <. A , C >. ) )
9	6, 8	eqeq12d 2208	. . . 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 2382	. . . . 5 |- ( ( ph /\ a = A ) -> ( a =/= b <-> A =/= b ) )
14	12	opeq1d 3810	. . . . . . . 8 |- ( ( ph /\ a = A ) -> <. a , C >. = <. A , C >. )
15	14	oveq2d 5934	. . . . . . 7 |- ( ( ph /\ a = A ) -> ( S sSet <. a , C >. ) = ( S sSet <. A , C >. ) )
16	15	oveq1d 5933	. . . . . 6 |- ( ( ph /\ a = A ) -> ( ( S sSet <. a , C >. ) sSet <. b , D >. ) = ( ( S sSet <. A , C >. ) sSet <. b , D >. ) )
17	14	oveq2d 5934	. . . . . 6 |- ( ( ph /\ a = A ) -> ( ( S sSet <. b , D >. ) sSet <. a , C >. ) = ( ( S sSet <. b , D >. ) sSet <. A , C >. ) )
18	16, 17	eqeq12d 2208	. . . . 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 2763	. . . . . . 7 |- a e. _V
28		vex 2763	. . . . . . 7 |- b e. _V
29	27, 28	setscom 12658	. . . . . 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 1250	. . . . 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 2812	. . 3 |- ( ph -> ( A =/= b -> ( ( S sSet <. A , C >. ) sSet <. b , D >. ) = ( ( S sSet <. b , D >. ) sSet <. A , C >. ) ) )
33	2, 10, 32	vtocld 2812	. 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 1364   e. wcel 2164   =/= wne 2364   <.cop 3621  (class class class)co 5918   sSet csts 12616

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-res 4671  df-iota 5215  df-fun 5256  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-sets 12625

This theorem is referenced by:  mgpress  13427