Theorem setscomd 12917

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 2396	. . . 4 ⊢ ((𝜑 ∧ 𝑏 = 𝐵) → (𝐴 ≠ 𝑏 ↔ 𝐴 ≠ 𝐵))
5	3	opeq1d 3827	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 = 𝐵) → ⟨𝑏, 𝐷⟩ = ⟨𝐵, 𝐷⟩)
6	5	oveq2d 5967	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 = 𝐵) → ((𝑆 sSet ⟨𝐴, 𝐶⟩) sSet ⟨𝑏, 𝐷⟩) = ((𝑆 sSet ⟨𝐴, 𝐶⟩) sSet ⟨𝐵, 𝐷⟩))
7	5	oveq2d 5967	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 = 𝐵) → (𝑆 sSet ⟨𝑏, 𝐷⟩) = (𝑆 sSet ⟨𝐵, 𝐷⟩))
8	7	oveq1d 5966	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 = 𝐵) → ((𝑆 sSet ⟨𝑏, 𝐷⟩) sSet ⟨𝐴, 𝐶⟩) = ((𝑆 sSet ⟨𝐵, 𝐷⟩) sSet ⟨𝐴, 𝐶⟩))
9	6, 8	eqeq12d 2221	. . . 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 2395	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 = 𝐴) → (𝑎 ≠ 𝑏 ↔ 𝐴 ≠ 𝑏))
14	12	opeq1d 3827	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 = 𝐴) → ⟨𝑎, 𝐶⟩ = ⟨𝐴, 𝐶⟩)
15	14	oveq2d 5967	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 = 𝐴) → (𝑆 sSet ⟨𝑎, 𝐶⟩) = (𝑆 sSet ⟨𝐴, 𝐶⟩))
16	15	oveq1d 5966	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 = 𝐴) → ((𝑆 sSet ⟨𝑎, 𝐶⟩) sSet ⟨𝑏, 𝐷⟩) = ((𝑆 sSet ⟨𝐴, 𝐶⟩) sSet ⟨𝑏, 𝐷⟩))
17	14	oveq2d 5967	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 = 𝐴) → ((𝑆 sSet ⟨𝑏, 𝐷⟩) sSet ⟨𝑎, 𝐶⟩) = ((𝑆 sSet ⟨𝑏, 𝐷⟩) sSet ⟨𝐴, 𝐶⟩))
18	16, 17	eqeq12d 2221	. . . . 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 2776	. . . . . . 7 ⊢ 𝑎 ∈ V
28		vex 2776	. . . . . . 7 ⊢ 𝑏 ∈ V
29	27, 28	setscom 12916	. . . . . 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 1251	. . . . 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 2826	. . 3 ⊢ (𝜑 → (𝐴 ≠ 𝑏 → ((𝑆 sSet ⟨𝐴, 𝐶⟩) sSet ⟨𝑏, 𝐷⟩) = ((𝑆 sSet ⟨𝑏, 𝐷⟩) sSet ⟨𝐴, 𝐶⟩)))
33	2, 10, 32	vtocld 2826	. 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 1373   ∈ wcel 2177   ≠ wne 2377  ⟨cop 3637  (class class class)co 5951   sSet csts 12874

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4166  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-setind 4589

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-sbc 3000  df-dif 3169  df-un 3171  df-in 3173  df-ss 3180  df-nul 3462  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-br 4048  df-opab 4110  df-id 4344  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-res 4691  df-iota 5237  df-fun 5278  df-fv 5284  df-ov 5954  df-oprab 5955  df-mpo 5956  df-sets 12883

This theorem is referenced by:  mgpress  13737