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Theorem prel12gOLD 4519
Description: Obsolete version of prel12g 4531 as of 12-Jun-2022. (Contributed by AV, 9-Dec-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
prel12gOLD (((𝐴𝑉𝐵𝑊) ∧ (𝐶𝑋𝐷𝑌)) → (¬ 𝐴 = 𝐵 → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 ∈ {𝐶, 𝐷} ∧ 𝐵 ∈ {𝐶, 𝐷}))))

Proof of Theorem prel12gOLD
Dummy variables 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq1 2775 . . . . . . 7 (𝑥 = 𝐴 → (𝑥 = 𝑦𝐴 = 𝑦))
21notbid 307 . . . . . 6 (𝑥 = 𝐴 → (¬ 𝑥 = 𝑦 ↔ ¬ 𝐴 = 𝑦))
3 preq1 4405 . . . . . . . 8 (𝑥 = 𝐴 → {𝑥, 𝑦} = {𝐴, 𝑦})
43eqeq1d 2773 . . . . . . 7 (𝑥 = 𝐴 → ({𝑥, 𝑦} = {𝑧, 𝐷} ↔ {𝐴, 𝑦} = {𝑧, 𝐷}))
5 eleq1 2838 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥 ∈ {𝑧, 𝐷} ↔ 𝐴 ∈ {𝑧, 𝐷}))
65anbi1d 615 . . . . . . 7 (𝑥 = 𝐴 → ((𝑥 ∈ {𝑧, 𝐷} ∧ 𝑦 ∈ {𝑧, 𝐷}) ↔ (𝐴 ∈ {𝑧, 𝐷} ∧ 𝑦 ∈ {𝑧, 𝐷})))
74, 6bibi12d 334 . . . . . 6 (𝑥 = 𝐴 → (({𝑥, 𝑦} = {𝑧, 𝐷} ↔ (𝑥 ∈ {𝑧, 𝐷} ∧ 𝑦 ∈ {𝑧, 𝐷})) ↔ ({𝐴, 𝑦} = {𝑧, 𝐷} ↔ (𝐴 ∈ {𝑧, 𝐷} ∧ 𝑦 ∈ {𝑧, 𝐷}))))
82, 7imbi12d 333 . . . . 5 (𝑥 = 𝐴 → ((¬ 𝑥 = 𝑦 → ({𝑥, 𝑦} = {𝑧, 𝐷} ↔ (𝑥 ∈ {𝑧, 𝐷} ∧ 𝑦 ∈ {𝑧, 𝐷}))) ↔ (¬ 𝐴 = 𝑦 → ({𝐴, 𝑦} = {𝑧, 𝐷} ↔ (𝐴 ∈ {𝑧, 𝐷} ∧ 𝑦 ∈ {𝑧, 𝐷})))))
98imbi2d 329 . . . 4 (𝑥 = 𝐴 → ((𝐷𝑌 → (¬ 𝑥 = 𝑦 → ({𝑥, 𝑦} = {𝑧, 𝐷} ↔ (𝑥 ∈ {𝑧, 𝐷} ∧ 𝑦 ∈ {𝑧, 𝐷})))) ↔ (𝐷𝑌 → (¬ 𝐴 = 𝑦 → ({𝐴, 𝑦} = {𝑧, 𝐷} ↔ (𝐴 ∈ {𝑧, 𝐷} ∧ 𝑦 ∈ {𝑧, 𝐷}))))))
10 eqeq2 2782 . . . . . . 7 (𝑦 = 𝐵 → (𝐴 = 𝑦𝐴 = 𝐵))
1110notbid 307 . . . . . 6 (𝑦 = 𝐵 → (¬ 𝐴 = 𝑦 ↔ ¬ 𝐴 = 𝐵))
12 preq2 4406 . . . . . . . 8 (𝑦 = 𝐵 → {𝐴, 𝑦} = {𝐴, 𝐵})
1312eqeq1d 2773 . . . . . . 7 (𝑦 = 𝐵 → ({𝐴, 𝑦} = {𝑧, 𝐷} ↔ {𝐴, 𝐵} = {𝑧, 𝐷}))
14 eleq1 2838 . . . . . . . 8 (𝑦 = 𝐵 → (𝑦 ∈ {𝑧, 𝐷} ↔ 𝐵 ∈ {𝑧, 𝐷}))
1514anbi2d 614 . . . . . . 7 (𝑦 = 𝐵 → ((𝐴 ∈ {𝑧, 𝐷} ∧ 𝑦 ∈ {𝑧, 𝐷}) ↔ (𝐴 ∈ {𝑧, 𝐷} ∧ 𝐵 ∈ {𝑧, 𝐷})))
1613, 15bibi12d 334 . . . . . 6 (𝑦 = 𝐵 → (({𝐴, 𝑦} = {𝑧, 𝐷} ↔ (𝐴 ∈ {𝑧, 𝐷} ∧ 𝑦 ∈ {𝑧, 𝐷})) ↔ ({𝐴, 𝐵} = {𝑧, 𝐷} ↔ (𝐴 ∈ {𝑧, 𝐷} ∧ 𝐵 ∈ {𝑧, 𝐷}))))
1711, 16imbi12d 333 . . . . 5 (𝑦 = 𝐵 → ((¬ 𝐴 = 𝑦 → ({𝐴, 𝑦} = {𝑧, 𝐷} ↔ (𝐴 ∈ {𝑧, 𝐷} ∧ 𝑦 ∈ {𝑧, 𝐷}))) ↔ (¬ 𝐴 = 𝐵 → ({𝐴, 𝐵} = {𝑧, 𝐷} ↔ (𝐴 ∈ {𝑧, 𝐷} ∧ 𝐵 ∈ {𝑧, 𝐷})))))
1817imbi2d 329 . . . 4 (𝑦 = 𝐵 → ((𝐷𝑌 → (¬ 𝐴 = 𝑦 → ({𝐴, 𝑦} = {𝑧, 𝐷} ↔ (𝐴 ∈ {𝑧, 𝐷} ∧ 𝑦 ∈ {𝑧, 𝐷})))) ↔ (𝐷𝑌 → (¬ 𝐴 = 𝐵 → ({𝐴, 𝐵} = {𝑧, 𝐷} ↔ (𝐴 ∈ {𝑧, 𝐷} ∧ 𝐵 ∈ {𝑧, 𝐷}))))))
19 preq1 4405 . . . . . . . 8 (𝑧 = 𝐶 → {𝑧, 𝐷} = {𝐶, 𝐷})
2019eqeq2d 2781 . . . . . . 7 (𝑧 = 𝐶 → ({𝐴, 𝐵} = {𝑧, 𝐷} ↔ {𝐴, 𝐵} = {𝐶, 𝐷}))
2119eleq2d 2836 . . . . . . . 8 (𝑧 = 𝐶 → (𝐴 ∈ {𝑧, 𝐷} ↔ 𝐴 ∈ {𝐶, 𝐷}))
2219eleq2d 2836 . . . . . . . 8 (𝑧 = 𝐶 → (𝐵 ∈ {𝑧, 𝐷} ↔ 𝐵 ∈ {𝐶, 𝐷}))
2321, 22anbi12d 616 . . . . . . 7 (𝑧 = 𝐶 → ((𝐴 ∈ {𝑧, 𝐷} ∧ 𝐵 ∈ {𝑧, 𝐷}) ↔ (𝐴 ∈ {𝐶, 𝐷} ∧ 𝐵 ∈ {𝐶, 𝐷})))
2420, 23bibi12d 334 . . . . . 6 (𝑧 = 𝐶 → (({𝐴, 𝐵} = {𝑧, 𝐷} ↔ (𝐴 ∈ {𝑧, 𝐷} ∧ 𝐵 ∈ {𝑧, 𝐷})) ↔ ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 ∈ {𝐶, 𝐷} ∧ 𝐵 ∈ {𝐶, 𝐷}))))
2524imbi2d 329 . . . . 5 (𝑧 = 𝐶 → ((¬ 𝐴 = 𝐵 → ({𝐴, 𝐵} = {𝑧, 𝐷} ↔ (𝐴 ∈ {𝑧, 𝐷} ∧ 𝐵 ∈ {𝑧, 𝐷}))) ↔ (¬ 𝐴 = 𝐵 → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 ∈ {𝐶, 𝐷} ∧ 𝐵 ∈ {𝐶, 𝐷})))))
2625imbi2d 329 . . . 4 (𝑧 = 𝐶 → ((𝐷𝑌 → (¬ 𝐴 = 𝐵 → ({𝐴, 𝐵} = {𝑧, 𝐷} ↔ (𝐴 ∈ {𝑧, 𝐷} ∧ 𝐵 ∈ {𝑧, 𝐷})))) ↔ (𝐷𝑌 → (¬ 𝐴 = 𝐵 → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 ∈ {𝐶, 𝐷} ∧ 𝐵 ∈ {𝐶, 𝐷}))))))
27 preq2 4406 . . . . . . . . 9 (𝑤 = 𝐷 → {𝑧, 𝑤} = {𝑧, 𝐷})
2827eqeq2d 2781 . . . . . . . 8 (𝑤 = 𝐷 → ({𝑥, 𝑦} = {𝑧, 𝑤} ↔ {𝑥, 𝑦} = {𝑧, 𝐷}))
2927eleq2d 2836 . . . . . . . . 9 (𝑤 = 𝐷 → (𝑥 ∈ {𝑧, 𝑤} ↔ 𝑥 ∈ {𝑧, 𝐷}))
3027eleq2d 2836 . . . . . . . . 9 (𝑤 = 𝐷 → (𝑦 ∈ {𝑧, 𝑤} ↔ 𝑦 ∈ {𝑧, 𝐷}))
3129, 30anbi12d 616 . . . . . . . 8 (𝑤 = 𝐷 → ((𝑥 ∈ {𝑧, 𝑤} ∧ 𝑦 ∈ {𝑧, 𝑤}) ↔ (𝑥 ∈ {𝑧, 𝐷} ∧ 𝑦 ∈ {𝑧, 𝐷})))
3228, 31bibi12d 334 . . . . . . 7 (𝑤 = 𝐷 → (({𝑥, 𝑦} = {𝑧, 𝑤} ↔ (𝑥 ∈ {𝑧, 𝑤} ∧ 𝑦 ∈ {𝑧, 𝑤})) ↔ ({𝑥, 𝑦} = {𝑧, 𝐷} ↔ (𝑥 ∈ {𝑧, 𝐷} ∧ 𝑦 ∈ {𝑧, 𝐷}))))
3332imbi2d 329 . . . . . 6 (𝑤 = 𝐷 → ((¬ 𝑥 = 𝑦 → ({𝑥, 𝑦} = {𝑧, 𝑤} ↔ (𝑥 ∈ {𝑧, 𝑤} ∧ 𝑦 ∈ {𝑧, 𝑤}))) ↔ (¬ 𝑥 = 𝑦 → ({𝑥, 𝑦} = {𝑧, 𝐷} ↔ (𝑥 ∈ {𝑧, 𝐷} ∧ 𝑦 ∈ {𝑧, 𝐷})))))
34 vex 3354 . . . . . . 7 𝑥 ∈ V
35 vex 3354 . . . . . . 7 𝑦 ∈ V
36 vex 3354 . . . . . . 7 𝑧 ∈ V
37 vex 3354 . . . . . . 7 𝑤 ∈ V
3834, 35, 36, 37prel12OLD 4515 . . . . . 6 𝑥 = 𝑦 → ({𝑥, 𝑦} = {𝑧, 𝑤} ↔ (𝑥 ∈ {𝑧, 𝑤} ∧ 𝑦 ∈ {𝑧, 𝑤})))
3933, 38vtoclg 3417 . . . . 5 (𝐷𝑌 → (¬ 𝑥 = 𝑦 → ({𝑥, 𝑦} = {𝑧, 𝐷} ↔ (𝑥 ∈ {𝑧, 𝐷} ∧ 𝑦 ∈ {𝑧, 𝐷}))))
4039a1i 11 . . . 4 ((𝑥𝑉𝑦𝑊𝑧𝑋) → (𝐷𝑌 → (¬ 𝑥 = 𝑦 → ({𝑥, 𝑦} = {𝑧, 𝐷} ↔ (𝑥 ∈ {𝑧, 𝐷} ∧ 𝑦 ∈ {𝑧, 𝐷})))))
419, 18, 26, 40vtocl3ga 3427 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐷𝑌 → (¬ 𝐴 = 𝐵 → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 ∈ {𝐶, 𝐷} ∧ 𝐵 ∈ {𝐶, 𝐷})))))
42413expa 1111 . 2 (((𝐴𝑉𝐵𝑊) ∧ 𝐶𝑋) → (𝐷𝑌 → (¬ 𝐴 = 𝐵 → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 ∈ {𝐶, 𝐷} ∧ 𝐵 ∈ {𝐶, 𝐷})))))
4342impr 442 1 (((𝐴𝑉𝐵𝑊) ∧ (𝐶𝑋𝐷𝑌)) → (¬ 𝐴 = 𝐵 → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 ∈ {𝐶, 𝐷} ∧ 𝐵 ∈ {𝐶, 𝐷}))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  {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-v 3353  df-un 3728  df-sn 4318  df-pr 4320
This theorem is referenced by: (None)
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