Theorem setscomd 12505

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 2366	. . . 4 ⊢ ((𝜑 ∧ 𝑏 = 𝐵) → (𝐴 ≠ 𝑏 ↔ 𝐴 ≠ 𝐵))
5	3	opeq1d 3786	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 = 𝐵) → ⟨𝑏, 𝐷⟩ = ⟨𝐵, 𝐷⟩)
6	5	oveq2d 5893	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 = 𝐵) → ((𝑆 sSet ⟨𝐴, 𝐶⟩) sSet ⟨𝑏, 𝐷⟩) = ((𝑆 sSet ⟨𝐴, 𝐶⟩) sSet ⟨𝐵, 𝐷⟩))
7	5	oveq2d 5893	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 = 𝐵) → (𝑆 sSet ⟨𝑏, 𝐷⟩) = (𝑆 sSet ⟨𝐵, 𝐷⟩))
8	7	oveq1d 5892	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 = 𝐵) → ((𝑆 sSet ⟨𝑏, 𝐷⟩) sSet ⟨𝐴, 𝐶⟩) = ((𝑆 sSet ⟨𝐵, 𝐷⟩) sSet ⟨𝐴, 𝐶⟩))
9	6, 8	eqeq12d 2192	. . . 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 2365	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 = 𝐴) → (𝑎 ≠ 𝑏 ↔ 𝐴 ≠ 𝑏))
14	12	opeq1d 3786	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 = 𝐴) → ⟨𝑎, 𝐶⟩ = ⟨𝐴, 𝐶⟩)
15	14	oveq2d 5893	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 = 𝐴) → (𝑆 sSet ⟨𝑎, 𝐶⟩) = (𝑆 sSet ⟨𝐴, 𝐶⟩))
16	15	oveq1d 5892	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 = 𝐴) → ((𝑆 sSet ⟨𝑎, 𝐶⟩) sSet ⟨𝑏, 𝐷⟩) = ((𝑆 sSet ⟨𝐴, 𝐶⟩) sSet ⟨𝑏, 𝐷⟩))
17	14	oveq2d 5893	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 = 𝐴) → ((𝑆 sSet ⟨𝑏, 𝐷⟩) sSet ⟨𝑎, 𝐶⟩) = ((𝑆 sSet ⟨𝑏, 𝐷⟩) sSet ⟨𝐴, 𝐶⟩))
18	16, 17	eqeq12d 2192	. . . . 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 2742	. . . . . . 7 ⊢ 𝑎 ∈ V
28		vex 2742	. . . . . . 7 ⊢ 𝑏 ∈ V
29	27, 28	setscom 12504	. . . . . 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 1239	. . . . 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 2791	. . 3 ⊢ (𝜑 → (𝐴 ≠ 𝑏 → ((𝑆 sSet ⟨𝐴, 𝐶⟩) sSet ⟨𝑏, 𝐷⟩) = ((𝑆 sSet ⟨𝑏, 𝐷⟩) sSet ⟨𝐴, 𝐶⟩)))
33	2, 10, 32	vtocld 2791	. 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 1353   ∈ wcel 2148   ≠ wne 2347  ⟨cop 3597  (class class class)co 5877   sSet csts 12462

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-res 4640  df-iota 5180  df-fun 5220  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-sets 12471

This theorem is referenced by:  mgpress  13146