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Theorem prel12g 4792
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 4791 . 2 ((𝐴𝑉𝐵𝑊𝐴𝐵) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
2 prid1g 4694 . . . . . . . . 9 (𝐴𝑉𝐴 ∈ {𝐴, 𝐷})
323ad2ant1 1127 . . . . . . . 8 ((𝐴𝑉𝐵𝑊𝐴𝐵) → 𝐴 ∈ {𝐴, 𝐷})
43adantr 481 . . . . . . 7 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ 𝐴 = 𝐶) → 𝐴 ∈ {𝐴, 𝐷})
5 preq1 4667 . . . . . . . 8 (𝐴 = 𝐶 → {𝐴, 𝐷} = {𝐶, 𝐷})
65adantl 482 . . . . . . 7 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ 𝐴 = 𝐶) → {𝐴, 𝐷} = {𝐶, 𝐷})
74, 6eleqtrd 2919 . . . . . 6 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ 𝐴 = 𝐶) → 𝐴 ∈ {𝐶, 𝐷})
87ex 413 . . . . 5 ((𝐴𝑉𝐵𝑊𝐴𝐵) → (𝐴 = 𝐶𝐴 ∈ {𝐶, 𝐷}))
9 prid2g 4695 . . . . . . 7 (𝐵𝑊𝐵 ∈ {𝐶, 𝐵})
1093ad2ant2 1128 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐴𝐵) → 𝐵 ∈ {𝐶, 𝐵})
11 preq2 4668 . . . . . . 7 (𝐵 = 𝐷 → {𝐶, 𝐵} = {𝐶, 𝐷})
1211eleq2d 2902 . . . . . 6 (𝐵 = 𝐷 → (𝐵 ∈ {𝐶, 𝐵} ↔ 𝐵 ∈ {𝐶, 𝐷}))
1310, 12syl5ibcom 246 . . . . 5 ((𝐴𝑉𝐵𝑊𝐴𝐵) → (𝐵 = 𝐷𝐵 ∈ {𝐶, 𝐷}))
148, 13anim12d 608 . . . 4 ((𝐴𝑉𝐵𝑊𝐴𝐵) → ((𝐴 = 𝐶𝐵 = 𝐷) → (𝐴 ∈ {𝐶, 𝐷} ∧ 𝐵 ∈ {𝐶, 𝐷})))
15 prid2g 4695 . . . . . . 7 (𝐴𝑉𝐴 ∈ {𝐶, 𝐴})
16153ad2ant1 1127 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐴𝐵) → 𝐴 ∈ {𝐶, 𝐴})
17 preq2 4668 . . . . . . 7 (𝐴 = 𝐷 → {𝐶, 𝐴} = {𝐶, 𝐷})
1817eleq2d 2902 . . . . . 6 (𝐴 = 𝐷 → (𝐴 ∈ {𝐶, 𝐴} ↔ 𝐴 ∈ {𝐶, 𝐷}))
1916, 18syl5ibcom 246 . . . . 5 ((𝐴𝑉𝐵𝑊𝐴𝐵) → (𝐴 = 𝐷𝐴 ∈ {𝐶, 𝐷}))
20 prid1g 4694 . . . . . . 7 (𝐵𝑊𝐵 ∈ {𝐵, 𝐷})
21203ad2ant2 1128 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐴𝐵) → 𝐵 ∈ {𝐵, 𝐷})
22 preq1 4667 . . . . . . 7 (𝐵 = 𝐶 → {𝐵, 𝐷} = {𝐶, 𝐷})
2322eleq2d 2902 . . . . . 6 (𝐵 = 𝐶 → (𝐵 ∈ {𝐵, 𝐷} ↔ 𝐵 ∈ {𝐶, 𝐷}))
2421, 23syl5ibcom 246 . . . . 5 ((𝐴𝑉𝐵𝑊𝐴𝐵) → (𝐵 = 𝐶𝐵 ∈ {𝐶, 𝐷}))
2519, 24anim12d 608 . . . 4 ((𝐴𝑉𝐵𝑊𝐴𝐵) → ((𝐴 = 𝐷𝐵 = 𝐶) → (𝐴 ∈ {𝐶, 𝐷} ∧ 𝐵 ∈ {𝐶, 𝐷})))
2614, 25jaod 855 . . 3 ((𝐴𝑉𝐵𝑊𝐴𝐵) → (((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)) → (𝐴 ∈ {𝐶, 𝐷} ∧ 𝐵 ∈ {𝐶, 𝐷})))
27 elprg 4584 . . . . . 6 (𝐴𝑉 → (𝐴 ∈ {𝐶, 𝐷} ↔ (𝐴 = 𝐶𝐴 = 𝐷)))
28273ad2ant1 1127 . . . . 5 ((𝐴𝑉𝐵𝑊𝐴𝐵) → (𝐴 ∈ {𝐶, 𝐷} ↔ (𝐴 = 𝐶𝐴 = 𝐷)))
29 elprg 4584 . . . . . 6 (𝐵𝑊 → (𝐵 ∈ {𝐶, 𝐷} ↔ (𝐵 = 𝐶𝐵 = 𝐷)))
30293ad2ant2 1128 . . . . 5 ((𝐴𝑉𝐵𝑊𝐴𝐵) → (𝐵 ∈ {𝐶, 𝐷} ↔ (𝐵 = 𝐶𝐵 = 𝐷)))
3128, 30anbi12d 630 . . . 4 ((𝐴𝑉𝐵𝑊𝐴𝐵) → ((𝐴 ∈ {𝐶, 𝐷} ∧ 𝐵 ∈ {𝐶, 𝐷}) ↔ ((𝐴 = 𝐶𝐴 = 𝐷) ∧ (𝐵 = 𝐶𝐵 = 𝐷))))
32 eqtr3 2847 . . . . . . . 8 ((𝐴 = 𝐶𝐵 = 𝐶) → 𝐴 = 𝐵)
33 eqneqall 3031 . . . . . . . 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 864 . . . . . . . 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 863 . . . . . . . 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 2847 . . . . . . . 8 ((𝐴 = 𝐷𝐵 = 𝐷) → 𝐴 = 𝐵)
4039, 33syl 17 . . . . . . 7 ((𝐴 = 𝐷𝐵 = 𝐷) → (𝐴𝐵 → ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
4134, 36, 38, 40ccase 1031 . . . . . 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 241 . . 3 ((𝐴𝑉𝐵𝑊𝐴𝐵) → ((𝐴 ∈ {𝐶, 𝐷} ∧ 𝐵 ∈ {𝐶, 𝐷}) → ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
4526, 44impbid 213 . 2 ((𝐴𝑉𝐵𝑊𝐴𝐵) → (((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)) ↔ (𝐴 ∈ {𝐶, 𝐷} ∧ 𝐵 ∈ {𝐶, 𝐷})))
461, 45bitrd 280 1 ((𝐴𝑉𝐵𝑊𝐴𝐵) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 ∈ {𝐶, 𝐷} ∧ 𝐵 ∈ {𝐶, 𝐷})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wo 843  w3a 1081   = wceq 1530  wcel 2107  wne 3020  {cpr 4565
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-v 3501  df-dif 3942  df-un 3944  df-nul 4295  df-sn 4564  df-pr 4566
This theorem is referenced by:  dfac2b  9548  hash2prd  13826
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