Theorem setscomd 12521

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

Hypotheses

Ref	Expression
setscomd.a	⊢ (𝜑 → 𝐴 ∈ 𝑌)
setscomd.b	⊢ (𝜑 → 𝐵 ∈ 𝑍)
setscomd.s	⊢ (𝜑 → 𝑆 ∈ 𝑉)
setscomd.ab	⊢ (𝜑 → 𝐴 ≠ 𝐵)
setscomd.c	⊢ (𝜑 → 𝐶 ∈ 𝑊)
setscomd.d	⊢ (𝜑 → 𝐷 ∈ 𝑋)

Assertion

Ref	Expression
setscomd	⊢ (𝜑 → ((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 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
2		setscomd.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑍)
3		simpr 110	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
4	3	neeq2d 2379	. . . 4 ⊢ ((𝜑 ∧ 𝑏 = 𝐵) → (𝐴 ≠ 𝑏 ↔ 𝐴 ≠ 𝐵))
5	3	opeq1d 3799	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 = 𝐵) → ⟨𝑏, 𝐷⟩ = ⟨𝐵, 𝐷⟩)
6	5	oveq2d 5907	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 = 𝐵) → ((𝑆 sSet ⟨𝐴, 𝐶⟩) sSet ⟨𝑏, 𝐷⟩) = ((𝑆 sSet ⟨𝐴, 𝐶⟩) sSet ⟨𝐵, 𝐷⟩))
7	5	oveq2d 5907	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 = 𝐵) → (𝑆 sSet ⟨𝑏, 𝐷⟩) = (𝑆 sSet ⟨𝐵, 𝐷⟩))
8	7	oveq1d 5906	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 = 𝐵) → ((𝑆 sSet ⟨𝑏, 𝐷⟩) sSet ⟨𝐴, 𝐶⟩) = ((𝑆 sSet ⟨𝐵, 𝐷⟩) sSet ⟨𝐴, 𝐶⟩))
9	6, 8	eqeq12d 2204	. . . 4 ⊢ ((𝜑 ∧ 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 ⊢ ((𝜑 ∧ 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 ⊢ (𝜑 → 𝐴 ∈ 𝑌)
12		simpr 110	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 = 𝐴) → 𝑎 = 𝐴)
13	12	neeq1d 2378	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 = 𝐴) → (𝑎 ≠ 𝑏 ↔ 𝐴 ≠ 𝑏))
14	12	opeq1d 3799	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 = 𝐴) → ⟨𝑎, 𝐶⟩ = ⟨𝐴, 𝐶⟩)
15	14	oveq2d 5907	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 = 𝐴) → (𝑆 sSet ⟨𝑎, 𝐶⟩) = (𝑆 sSet ⟨𝐴, 𝐶⟩))
16	15	oveq1d 5906	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 = 𝐴) → ((𝑆 sSet ⟨𝑎, 𝐶⟩) sSet ⟨𝑏, 𝐷⟩) = ((𝑆 sSet ⟨𝐴, 𝐶⟩) sSet ⟨𝑏, 𝐷⟩))
17	14	oveq2d 5907	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 = 𝐴) → ((𝑆 sSet ⟨𝑏, 𝐷⟩) sSet ⟨𝑎, 𝐶⟩) = ((𝑆 sSet ⟨𝑏, 𝐷⟩) sSet ⟨𝐴, 𝐶⟩))
18	16, 17	eqeq12d 2204	. . . . 5 ⊢ ((𝜑 ∧ 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 ⊢ ((𝜑 ∧ 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 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
21	20	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ≠ 𝑏) → 𝑆 ∈ 𝑉)
22		simpr 110	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ≠ 𝑏) → 𝑎 ≠ 𝑏)
23		setscomd.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ 𝑊)
24	23	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ≠ 𝑏) → 𝐶 ∈ 𝑊)
25		setscomd.d	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ 𝑋)
26	25	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ≠ 𝑏) → 𝐷 ∈ 𝑋)
27		vex 2755	. . . . . . 7 ⊢ 𝑎 ∈ V
28		vex 2755	. . . . . . 7 ⊢ 𝑏 ∈ V
29	27, 28	setscom 12520	. . . . . 6 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑎 ≠ 𝑏) ∧ (𝐶 ∈ 𝑊 ∧ 𝐷 ∈ 𝑋)) → ((𝑆 sSet ⟨𝑎, 𝐶⟩) sSet ⟨𝑏, 𝐷⟩) = ((𝑆 sSet ⟨𝑏, 𝐷⟩) sSet ⟨𝑎, 𝐶⟩))
30	21, 22, 24, 26, 29	syl22anc 1250	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ≠ 𝑏) → ((𝑆 sSet ⟨𝑎, 𝐶⟩) sSet ⟨𝑏, 𝐷⟩) = ((𝑆 sSet ⟨𝑏, 𝐷⟩) sSet ⟨𝑎, 𝐶⟩))
31	30	ex 115	. . . 4 ⊢ (𝜑 → (𝑎 ≠ 𝑏 → ((𝑆 sSet ⟨𝑎, 𝐶⟩) sSet ⟨𝑏, 𝐷⟩) = ((𝑆 sSet ⟨𝑏, 𝐷⟩) sSet ⟨𝑎, 𝐶⟩)))
32	11, 19, 31	vtocld 2804	. . 3 ⊢ (𝜑 → (𝐴 ≠ 𝑏 → ((𝑆 sSet ⟨𝐴, 𝐶⟩) sSet ⟨𝑏, 𝐷⟩) = ((𝑆 sSet ⟨𝑏, 𝐷⟩) sSet ⟨𝐴, 𝐶⟩)))
33	2, 10, 32	vtocld 2804	. 2 ⊢ (𝜑 → (𝐴 ≠ 𝐵 → ((𝑆 sSet ⟨𝐴, 𝐶⟩) sSet ⟨𝐵, 𝐷⟩) = ((𝑆 sSet ⟨𝐵, 𝐷⟩) sSet ⟨𝐴, 𝐶⟩)))
34	1, 33	mpd 13	1 ⊢ (𝜑 → ((𝑆 sSet ⟨𝐴, 𝐶⟩) sSet ⟨𝐵, 𝐷⟩) = ((𝑆 sSet ⟨𝐵, 𝐷⟩) sSet ⟨𝐴, 𝐶⟩))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2160   ≠ wne 2360  ⟨cop 3610  (class class class)co 5891   sSet csts 12478

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 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551

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 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-res 4653  df-iota 5193  df-fun 5233  df-fv 5239  df-ov 5894  df-oprab 5895  df-mpo 5896  df-sets 12487

This theorem is referenced by:  mgpress  13246