Theorem setscomd 13371

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 2439	. . . 4 |- ( ( ph /\ b = B ) -> ( A =/= b <-> A =/= B ) )
5	3	opeq1d 3905	. . . . . 6 |- ( ( ph /\ b = B ) -> <. b , D >. = <. B , D >. )
6	5	oveq2d 6091	. . . . 5 |- ( ( ph /\ b = B ) -> ( ( S sSet <. A , C >. ) sSet <. b , D >. ) = ( ( S sSet <. A , C >. ) sSet <. B , D >. ) )
7	5	oveq2d 6091	. . . . . 6 |- ( ( ph /\ b = B ) -> ( S sSet <. b , D >. ) = ( S sSet <. B , D >. ) )
8	7	oveq1d 6090	. . . . 5 |- ( ( ph /\ b = B ) -> ( ( S sSet <. b , D >. ) sSet <. A , C >. ) = ( ( S sSet <. B , D >. ) sSet <. A , C >. ) )
9	6, 8	eqeq12d 2253	. . . 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 2438	. . . . 5 |- ( ( ph /\ a = A ) -> ( a =/= b <-> A =/= b ) )
14	12	opeq1d 3905	. . . . . . . 8 |- ( ( ph /\ a = A ) -> <. a , C >. = <. A , C >. )
15	14	oveq2d 6091	. . . . . . 7 |- ( ( ph /\ a = A ) -> ( S sSet <. a , C >. ) = ( S sSet <. A , C >. ) )
16	15	oveq1d 6090	. . . . . 6 |- ( ( ph /\ a = A ) -> ( ( S sSet <. a , C >. ) sSet <. b , D >. ) = ( ( S sSet <. A , C >. ) sSet <. b , D >. ) )
17	14	oveq2d 6091	. . . . . 6 |- ( ( ph /\ a = A ) -> ( ( S sSet <. b , D >. ) sSet <. a , C >. ) = ( ( S sSet <. b , D >. ) sSet <. A , C >. ) )
18	16, 17	eqeq12d 2253	. . . . 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 2824	. . . . . . 7 |- a e. _V
28		vex 2824	. . . . . . 7 |- b e. _V
29	27, 28	setscom 13370	. . . . . 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 1279	. . . . 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 2875	. . 3 |- ( ph -> ( A =/= b -> ( ( S sSet <. A , C >. ) sSet <. b , D >. ) = ( ( S sSet <. b , D >. ) sSet <. A , C >. ) ) )
33	2, 10, 32	vtocld 2875	. 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 1402   e. wcel 2209   =/= wne 2420   <.cop 3708  (class class class)co 6075   sSet csts 13328

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-res 4781  df-iota 5332  df-fun 5374  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-sets 13337

This theorem is referenced by:  mgpress  14205