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Theorem setscomd 12505
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 2366 . . . 4 ((𝜑𝑏 = 𝐵) → (𝐴𝑏𝐴𝐵))
53opeq1d 3786 . . . . . 6 ((𝜑𝑏 = 𝐵) → ⟨𝑏, 𝐷⟩ = ⟨𝐵, 𝐷⟩)
65oveq2d 5893 . . . . 5 ((𝜑𝑏 = 𝐵) → ((𝑆 sSet ⟨𝐴, 𝐶⟩) sSet ⟨𝑏, 𝐷⟩) = ((𝑆 sSet ⟨𝐴, 𝐶⟩) sSet ⟨𝐵, 𝐷⟩))
75oveq2d 5893 . . . . . 6 ((𝜑𝑏 = 𝐵) → (𝑆 sSet ⟨𝑏, 𝐷⟩) = (𝑆 sSet ⟨𝐵, 𝐷⟩))
87oveq1d 5892 . . . . 5 ((𝜑𝑏 = 𝐵) → ((𝑆 sSet ⟨𝑏, 𝐷⟩) sSet ⟨𝐴, 𝐶⟩) = ((𝑆 sSet ⟨𝐵, 𝐷⟩) sSet ⟨𝐴, 𝐶⟩))
96, 8eqeq12d 2192 . . . 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 2365 . . . . 5 ((𝜑𝑎 = 𝐴) → (𝑎𝑏𝐴𝑏))
1412opeq1d 3786 . . . . . . . 8 ((𝜑𝑎 = 𝐴) → ⟨𝑎, 𝐶⟩ = ⟨𝐴, 𝐶⟩)
1514oveq2d 5893 . . . . . . 7 ((𝜑𝑎 = 𝐴) → (𝑆 sSet ⟨𝑎, 𝐶⟩) = (𝑆 sSet ⟨𝐴, 𝐶⟩))
1615oveq1d 5892 . . . . . 6 ((𝜑𝑎 = 𝐴) → ((𝑆 sSet ⟨𝑎, 𝐶⟩) sSet ⟨𝑏, 𝐷⟩) = ((𝑆 sSet ⟨𝐴, 𝐶⟩) sSet ⟨𝑏, 𝐷⟩))
1714oveq2d 5893 . . . . . 6 ((𝜑𝑎 = 𝐴) → ((𝑆 sSet ⟨𝑏, 𝐷⟩) sSet ⟨𝑎, 𝐶⟩) = ((𝑆 sSet ⟨𝑏, 𝐷⟩) sSet ⟨𝐴, 𝐶⟩))
1816, 17eqeq12d 2192 . . . . 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 2742 . . . . . . 7 𝑎 ∈ V
28 vex 2742 . . . . . . 7 𝑏 ∈ V
2927, 28setscom 12504 . . . . . 6 (((𝑆𝑉𝑎𝑏) ∧ (𝐶𝑊𝐷𝑋)) → ((𝑆 sSet ⟨𝑎, 𝐶⟩) sSet ⟨𝑏, 𝐷⟩) = ((𝑆 sSet ⟨𝑏, 𝐷⟩) sSet ⟨𝑎, 𝐶⟩))
3021, 22, 24, 26, 29syl22anc 1239 . . . . 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 2791 . . 3 (𝜑 → (𝐴𝑏 → ((𝑆 sSet ⟨𝐴, 𝐶⟩) sSet ⟨𝑏, 𝐷⟩) = ((𝑆 sSet ⟨𝑏, 𝐷⟩) sSet ⟨𝐴, 𝐶⟩)))
332, 10, 32vtocld 2791 . 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 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
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