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Theorem prel12g 4531
Description: Equality of two unordered pairs. (Contributed by NM, 17-Oct-1996.) (Revised by AV, 9-Dec-2018.) (Revised by AV, 12-Jun-2022.)
Assertion
Ref Expression
prel12g ((𝐴𝑉𝐵𝑊𝐴𝐵) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 ∈ {𝐶, 𝐷} ∧ 𝐵 ∈ {𝐶, 𝐷})))

Proof of Theorem prel12g
StepHypRef Expression
1 preq12nebg 4530 . 2 ((𝐴𝑉𝐵𝑊𝐴𝐵) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
2 prid1g 4432 . . . . . . . . 9 (𝐴𝑉𝐴 ∈ {𝐴, 𝐷})
323ad2ant1 1127 . . . . . . . 8 ((𝐴𝑉𝐵𝑊𝐴𝐵) → 𝐴 ∈ {𝐴, 𝐷})
43adantr 466 . . . . . . 7 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ 𝐴 = 𝐶) → 𝐴 ∈ {𝐴, 𝐷})
5 preq1 4405 . . . . . . . 8 (𝐴 = 𝐶 → {𝐴, 𝐷} = {𝐶, 𝐷})
65adantl 467 . . . . . . 7 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ 𝐴 = 𝐶) → {𝐴, 𝐷} = {𝐶, 𝐷})
74, 6eleqtrd 2852 . . . . . 6 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ 𝐴 = 𝐶) → 𝐴 ∈ {𝐶, 𝐷})
87ex 397 . . . . 5 ((𝐴𝑉𝐵𝑊𝐴𝐵) → (𝐴 = 𝐶𝐴 ∈ {𝐶, 𝐷}))
9 prid2g 4433 . . . . . . 7 (𝐵𝑊𝐵 ∈ {𝐶, 𝐵})
1093ad2ant2 1128 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐴𝐵) → 𝐵 ∈ {𝐶, 𝐵})
11 preq2 4406 . . . . . . 7 (𝐵 = 𝐷 → {𝐶, 𝐵} = {𝐶, 𝐷})
1211eleq2d 2836 . . . . . 6 (𝐵 = 𝐷 → (𝐵 ∈ {𝐶, 𝐵} ↔ 𝐵 ∈ {𝐶, 𝐷}))
1310, 12syl5ibcom 235 . . . . 5 ((𝐴𝑉𝐵𝑊𝐴𝐵) → (𝐵 = 𝐷𝐵 ∈ {𝐶, 𝐷}))
148, 13anim12d 596 . . . 4 ((𝐴𝑉𝐵𝑊𝐴𝐵) → ((𝐴 = 𝐶𝐵 = 𝐷) → (𝐴 ∈ {𝐶, 𝐷} ∧ 𝐵 ∈ {𝐶, 𝐷})))
15 prid2g 4433 . . . . . . 7 (𝐴𝑉𝐴 ∈ {𝐶, 𝐴})
16153ad2ant1 1127 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐴𝐵) → 𝐴 ∈ {𝐶, 𝐴})
17 preq2 4406 . . . . . . 7 (𝐴 = 𝐷 → {𝐶, 𝐴} = {𝐶, 𝐷})
1817eleq2d 2836 . . . . . 6 (𝐴 = 𝐷 → (𝐴 ∈ {𝐶, 𝐴} ↔ 𝐴 ∈ {𝐶, 𝐷}))
1916, 18syl5ibcom 235 . . . . 5 ((𝐴𝑉𝐵𝑊𝐴𝐵) → (𝐴 = 𝐷𝐴 ∈ {𝐶, 𝐷}))
20 prid1g 4432 . . . . . . 7 (𝐵𝑊𝐵 ∈ {𝐵, 𝐷})
21203ad2ant2 1128 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐴𝐵) → 𝐵 ∈ {𝐵, 𝐷})
22 preq1 4405 . . . . . . 7 (𝐵 = 𝐶 → {𝐵, 𝐷} = {𝐶, 𝐷})
2322eleq2d 2836 . . . . . 6 (𝐵 = 𝐶 → (𝐵 ∈ {𝐵, 𝐷} ↔ 𝐵 ∈ {𝐶, 𝐷}))
2421, 23syl5ibcom 235 . . . . 5 ((𝐴𝑉𝐵𝑊𝐴𝐵) → (𝐵 = 𝐶𝐵 ∈ {𝐶, 𝐷}))
2519, 24anim12d 596 . . . 4 ((𝐴𝑉𝐵𝑊𝐴𝐵) → ((𝐴 = 𝐷𝐵 = 𝐶) → (𝐴 ∈ {𝐶, 𝐷} ∧ 𝐵 ∈ {𝐶, 𝐷})))
2614, 25jaod 848 . . 3 ((𝐴𝑉𝐵𝑊𝐴𝐵) → (((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)) → (𝐴 ∈ {𝐶, 𝐷} ∧ 𝐵 ∈ {𝐶, 𝐷})))
27 elprg 4337 . . . . . 6 (𝐴𝑉 → (𝐴 ∈ {𝐶, 𝐷} ↔ (𝐴 = 𝐶𝐴 = 𝐷)))
28273ad2ant1 1127 . . . . 5 ((𝐴𝑉𝐵𝑊𝐴𝐵) → (𝐴 ∈ {𝐶, 𝐷} ↔ (𝐴 = 𝐶𝐴 = 𝐷)))
29 elprg 4337 . . . . . 6 (𝐵𝑊 → (𝐵 ∈ {𝐶, 𝐷} ↔ (𝐵 = 𝐶𝐵 = 𝐷)))
30293ad2ant2 1128 . . . . 5 ((𝐴𝑉𝐵𝑊𝐴𝐵) → (𝐵 ∈ {𝐶, 𝐷} ↔ (𝐵 = 𝐶𝐵 = 𝐷)))
3128, 30anbi12d 616 . . . 4 ((𝐴𝑉𝐵𝑊𝐴𝐵) → ((𝐴 ∈ {𝐶, 𝐷} ∧ 𝐵 ∈ {𝐶, 𝐷}) ↔ ((𝐴 = 𝐶𝐴 = 𝐷) ∧ (𝐵 = 𝐶𝐵 = 𝐷))))
32 eqtr3 2792 . . . . . . . 8 ((𝐴 = 𝐶𝐵 = 𝐶) → 𝐴 = 𝐵)
33 eqneqall 2954 . . . . . . . 8 (𝐴 = 𝐵 → (𝐴𝐵 → ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
3432, 33syl 17 . . . . . . 7 ((𝐴 = 𝐶𝐵 = 𝐶) → (𝐴𝐵 → ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
35 olc 857 . . . . . . . 8 ((𝐴 = 𝐷𝐵 = 𝐶) → ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)))
3635a1d 25 . . . . . . 7 ((𝐴 = 𝐷𝐵 = 𝐶) → (𝐴𝐵 → ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
37 orc 856 . . . . . . . 8 ((𝐴 = 𝐶𝐵 = 𝐷) → ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)))
3837a1d 25 . . . . . . 7 ((𝐴 = 𝐶𝐵 = 𝐷) → (𝐴𝐵 → ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
39 eqtr3 2792 . . . . . . . 8 ((𝐴 = 𝐷𝐵 = 𝐷) → 𝐴 = 𝐵)
4039, 33syl 17 . . . . . . 7 ((𝐴 = 𝐷𝐵 = 𝐷) → (𝐴𝐵 → ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
4134, 36, 38, 40ccase 1022 . . . . . 6 (((𝐴 = 𝐶𝐴 = 𝐷) ∧ (𝐵 = 𝐶𝐵 = 𝐷)) → (𝐴𝐵 → ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
4241com12 32 . . . . 5 (𝐴𝐵 → (((𝐴 = 𝐶𝐴 = 𝐷) ∧ (𝐵 = 𝐶𝐵 = 𝐷)) → ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
43423ad2ant3 1129 . . . 4 ((𝐴𝑉𝐵𝑊𝐴𝐵) → (((𝐴 = 𝐶𝐴 = 𝐷) ∧ (𝐵 = 𝐶𝐵 = 𝐷)) → ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
4431, 43sylbid 230 . . 3 ((𝐴𝑉𝐵𝑊𝐴𝐵) → ((𝐴 ∈ {𝐶, 𝐷} ∧ 𝐵 ∈ {𝐶, 𝐷}) → ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
4526, 44impbid 202 . 2 ((𝐴𝑉𝐵𝑊𝐴𝐵) → (((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)) ↔ (𝐴 ∈ {𝐶, 𝐷} ∧ 𝐵 ∈ {𝐶, 𝐷})))
461, 45bitrd 268 1 ((𝐴𝑉𝐵𝑊𝐴𝐵) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 ∈ {𝐶, 𝐷} ∧ 𝐵 ∈ {𝐶, 𝐷})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  wo 836  w3a 1071   = wceq 1631  wcel 2145  wne 2943  {cpr 4319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-v 3353  df-dif 3726  df-un 3728  df-nul 4064  df-sn 4318  df-pr 4320
This theorem is referenced by:  dfac2b  9157  hash2prd  13459
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