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Theorem setscomd 12516
Description: Different components can be set in any order. (Contributed by Jim Kingdon, 20-Feb-2025.)
Hypotheses
Ref Expression
setscomd.a (𝜑𝐴𝑌)
setscomd.b (𝜑𝐵𝑍)
setscomd.s (𝜑𝑆𝑉)
setscomd.ab (𝜑𝐴𝐵)
setscomd.c (𝜑𝐶𝑊)
setscomd.d (𝜑𝐷𝑋)
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.
StepHypRef Expression
1 setscomd.ab . 2 (𝜑𝐴𝐵)
2 setscomd.b . . 3 (𝜑𝐵𝑍)
3 simpr 110 . . . . 5 ((𝜑𝑏 = 𝐵) → 𝑏 = 𝐵)
43neeq2d 2376 . . . 4 ((𝜑𝑏 = 𝐵) → (𝐴𝑏𝐴𝐵))
53opeq1d 3796 . . . . . 6 ((𝜑𝑏 = 𝐵) → ⟨𝑏, 𝐷⟩ = ⟨𝐵, 𝐷⟩)
65oveq2d 5904 . . . . 5 ((𝜑𝑏 = 𝐵) → ((𝑆 sSet ⟨𝐴, 𝐶⟩) sSet ⟨𝑏, 𝐷⟩) = ((𝑆 sSet ⟨𝐴, 𝐶⟩) sSet ⟨𝐵, 𝐷⟩))
75oveq2d 5904 . . . . . 6 ((𝜑𝑏 = 𝐵) → (𝑆 sSet ⟨𝑏, 𝐷⟩) = (𝑆 sSet ⟨𝐵, 𝐷⟩))
87oveq1d 5903 . . . . 5 ((𝜑𝑏 = 𝐵) → ((𝑆 sSet ⟨𝑏, 𝐷⟩) sSet ⟨𝐴, 𝐶⟩) = ((𝑆 sSet ⟨𝐵, 𝐷⟩) sSet ⟨𝐴, 𝐶⟩))
96, 8eqeq12d 2202 . . . 4 ((𝜑𝑏 = 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⟩)))
104, 9imbi12d 234 . . 3 ((𝜑𝑏 = 𝐵) → ((𝐴𝑏 → ((𝑆 sSet ⟨𝐴, 𝐶⟩) sSet ⟨𝑏, 𝐷⟩) = ((𝑆 sSet ⟨𝑏, 𝐷⟩) sSet ⟨𝐴, 𝐶⟩)) ↔ (𝐴𝐵 → ((𝑆 sSet ⟨𝐴, 𝐶⟩) sSet ⟨𝐵, 𝐷⟩) = ((𝑆 sSet ⟨𝐵, 𝐷⟩) sSet ⟨𝐴, 𝐶⟩))))
11 setscomd.a . . . 4 (𝜑𝐴𝑌)
12 simpr 110 . . . . . 6 ((𝜑𝑎 = 𝐴) → 𝑎 = 𝐴)
1312neeq1d 2375 . . . . 5 ((𝜑𝑎 = 𝐴) → (𝑎𝑏𝐴𝑏))
1412opeq1d 3796 . . . . . . . 8 ((𝜑𝑎 = 𝐴) → ⟨𝑎, 𝐶⟩ = ⟨𝐴, 𝐶⟩)
1514oveq2d 5904 . . . . . . 7 ((𝜑𝑎 = 𝐴) → (𝑆 sSet ⟨𝑎, 𝐶⟩) = (𝑆 sSet ⟨𝐴, 𝐶⟩))
1615oveq1d 5903 . . . . . 6 ((𝜑𝑎 = 𝐴) → ((𝑆 sSet ⟨𝑎, 𝐶⟩) sSet ⟨𝑏, 𝐷⟩) = ((𝑆 sSet ⟨𝐴, 𝐶⟩) sSet ⟨𝑏, 𝐷⟩))
1714oveq2d 5904 . . . . . 6 ((𝜑𝑎 = 𝐴) → ((𝑆 sSet ⟨𝑏, 𝐷⟩) sSet ⟨𝑎, 𝐶⟩) = ((𝑆 sSet ⟨𝑏, 𝐷⟩) sSet ⟨𝐴, 𝐶⟩))
1816, 17eqeq12d 2202 . . . . 5 ((𝜑𝑎 = 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⟩)))
1913, 18imbi12d 234 . . . 4 ((𝜑𝑎 = 𝐴) → ((𝑎𝑏 → ((𝑆 sSet ⟨𝑎, 𝐶⟩) sSet ⟨𝑏, 𝐷⟩) = ((𝑆 sSet ⟨𝑏, 𝐷⟩) sSet ⟨𝑎, 𝐶⟩)) ↔ (𝐴𝑏 → ((𝑆 sSet ⟨𝐴, 𝐶⟩) sSet ⟨𝑏, 𝐷⟩) = ((𝑆 sSet ⟨𝑏, 𝐷⟩) sSet ⟨𝐴, 𝐶⟩))))
20 setscomd.s . . . . . . 7 (𝜑𝑆𝑉)
2120adantr 276 . . . . . 6 ((𝜑𝑎𝑏) → 𝑆𝑉)
22 simpr 110 . . . . . 6 ((𝜑𝑎𝑏) → 𝑎𝑏)
23 setscomd.c . . . . . . 7 (𝜑𝐶𝑊)
2423adantr 276 . . . . . 6 ((𝜑𝑎𝑏) → 𝐶𝑊)
25 setscomd.d . . . . . . 7 (𝜑𝐷𝑋)
2625adantr 276 . . . . . 6 ((𝜑𝑎𝑏) → 𝐷𝑋)
27 vex 2752 . . . . . . 7 𝑎 ∈ V
28 vex 2752 . . . . . . 7 𝑏 ∈ V
2927, 28setscom 12515 . . . . . 6 (((𝑆𝑉𝑎𝑏) ∧ (𝐶𝑊𝐷𝑋)) → ((𝑆 sSet ⟨𝑎, 𝐶⟩) sSet ⟨𝑏, 𝐷⟩) = ((𝑆 sSet ⟨𝑏, 𝐷⟩) sSet ⟨𝑎, 𝐶⟩))
3021, 22, 24, 26, 29syl22anc 1249 . . . . 5 ((𝜑𝑎𝑏) → ((𝑆 sSet ⟨𝑎, 𝐶⟩) sSet ⟨𝑏, 𝐷⟩) = ((𝑆 sSet ⟨𝑏, 𝐷⟩) sSet ⟨𝑎, 𝐶⟩))
3130ex 115 . . . 4 (𝜑 → (𝑎𝑏 → ((𝑆 sSet ⟨𝑎, 𝐶⟩) sSet ⟨𝑏, 𝐷⟩) = ((𝑆 sSet ⟨𝑏, 𝐷⟩) sSet ⟨𝑎, 𝐶⟩)))
3211, 19, 31vtocld 2801 . . 3 (𝜑 → (𝐴𝑏 → ((𝑆 sSet ⟨𝐴, 𝐶⟩) sSet ⟨𝑏, 𝐷⟩) = ((𝑆 sSet ⟨𝑏, 𝐷⟩) sSet ⟨𝐴, 𝐶⟩)))
332, 10, 32vtocld 2801 . 2 (𝜑 → (𝐴𝐵 → ((𝑆 sSet ⟨𝐴, 𝐶⟩) sSet ⟨𝐵, 𝐷⟩) = ((𝑆 sSet ⟨𝐵, 𝐷⟩) sSet ⟨𝐴, 𝐶⟩)))
341, 33mpd 13 1 (𝜑 → ((𝑆 sSet ⟨𝐴, 𝐶⟩) sSet ⟨𝐵, 𝐷⟩) = ((𝑆 sSet ⟨𝐵, 𝐷⟩) sSet ⟨𝐴, 𝐶⟩))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1363  wcel 2158  wne 2357  cop 3607  (class class class)co 5888   sSet csts 12473
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-sbc 2975  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-res 4650  df-iota 5190  df-fun 5230  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-sets 12482
This theorem is referenced by:  mgpress  13173
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