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Theorem preq12bg 3514
Description: Closed form of preq12b 3511. (Contributed by Scott Fenton, 28-Mar-2014.)
Assertion
Ref Expression
preq12bg (((A 𝑉 B 𝑊) (𝐶 𝑋 𝐷 𝑌)) → ({A, B} = {𝐶, 𝐷} ↔ ((A = 𝐶 B = 𝐷) (A = 𝐷 B = 𝐶))))

Proof of Theorem preq12bg
Dummy variables x y z w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 preq1 3417 . . . . . . 7 (x = A → {x, y} = {A, y})
21eqeq1d 2026 . . . . . 6 (x = A → ({x, y} = {z, 𝐷} ↔ {A, y} = {z, 𝐷}))
3 eqeq1 2024 . . . . . . . 8 (x = A → (x = zA = z))
43anbi1d 441 . . . . . . 7 (x = A → ((x = z y = 𝐷) ↔ (A = z y = 𝐷)))
5 eqeq1 2024 . . . . . . . 8 (x = A → (x = 𝐷A = 𝐷))
65anbi1d 441 . . . . . . 7 (x = A → ((x = 𝐷 y = z) ↔ (A = 𝐷 y = z)))
74, 6orbi12d 694 . . . . . 6 (x = A → (((x = z y = 𝐷) (x = 𝐷 y = z)) ↔ ((A = z y = 𝐷) (A = 𝐷 y = z))))
82, 7bibi12d 224 . . . . 5 (x = A → (({x, y} = {z, 𝐷} ↔ ((x = z y = 𝐷) (x = 𝐷 y = z))) ↔ ({A, y} = {z, 𝐷} ↔ ((A = z y = 𝐷) (A = 𝐷 y = z)))))
98imbi2d 219 . . . 4 (x = A → ((𝐷 𝑌 → ({x, y} = {z, 𝐷} ↔ ((x = z y = 𝐷) (x = 𝐷 y = z)))) ↔ (𝐷 𝑌 → ({A, y} = {z, 𝐷} ↔ ((A = z y = 𝐷) (A = 𝐷 y = z))))))
10 preq2 3418 . . . . . . 7 (y = B → {A, y} = {A, B})
1110eqeq1d 2026 . . . . . 6 (y = B → ({A, y} = {z, 𝐷} ↔ {A, B} = {z, 𝐷}))
12 eqeq1 2024 . . . . . . . 8 (y = B → (y = 𝐷B = 𝐷))
1312anbi2d 440 . . . . . . 7 (y = B → ((A = z y = 𝐷) ↔ (A = z B = 𝐷)))
14 eqeq1 2024 . . . . . . . 8 (y = B → (y = zB = z))
1514anbi2d 440 . . . . . . 7 (y = B → ((A = 𝐷 y = z) ↔ (A = 𝐷 B = z)))
1613, 15orbi12d 694 . . . . . 6 (y = B → (((A = z y = 𝐷) (A = 𝐷 y = z)) ↔ ((A = z B = 𝐷) (A = 𝐷 B = z))))
1711, 16bibi12d 224 . . . . 5 (y = B → (({A, y} = {z, 𝐷} ↔ ((A = z y = 𝐷) (A = 𝐷 y = z))) ↔ ({A, B} = {z, 𝐷} ↔ ((A = z B = 𝐷) (A = 𝐷 B = z)))))
1817imbi2d 219 . . . 4 (y = B → ((𝐷 𝑌 → ({A, y} = {z, 𝐷} ↔ ((A = z y = 𝐷) (A = 𝐷 y = z)))) ↔ (𝐷 𝑌 → ({A, B} = {z, 𝐷} ↔ ((A = z B = 𝐷) (A = 𝐷 B = z))))))
19 preq1 3417 . . . . . . 7 (z = 𝐶 → {z, 𝐷} = {𝐶, 𝐷})
2019eqeq2d 2029 . . . . . 6 (z = 𝐶 → ({A, B} = {z, 𝐷} ↔ {A, B} = {𝐶, 𝐷}))
21 eqeq2 2027 . . . . . . . 8 (z = 𝐶 → (A = zA = 𝐶))
2221anbi1d 441 . . . . . . 7 (z = 𝐶 → ((A = z B = 𝐷) ↔ (A = 𝐶 B = 𝐷)))
23 eqeq2 2027 . . . . . . . 8 (z = 𝐶 → (B = zB = 𝐶))
2423anbi2d 440 . . . . . . 7 (z = 𝐶 → ((A = 𝐷 B = z) ↔ (A = 𝐷 B = 𝐶)))
2522, 24orbi12d 694 . . . . . 6 (z = 𝐶 → (((A = z B = 𝐷) (A = 𝐷 B = z)) ↔ ((A = 𝐶 B = 𝐷) (A = 𝐷 B = 𝐶))))
2620, 25bibi12d 224 . . . . 5 (z = 𝐶 → (({A, B} = {z, 𝐷} ↔ ((A = z B = 𝐷) (A = 𝐷 B = z))) ↔ ({A, B} = {𝐶, 𝐷} ↔ ((A = 𝐶 B = 𝐷) (A = 𝐷 B = 𝐶)))))
2726imbi2d 219 . . . 4 (z = 𝐶 → ((𝐷 𝑌 → ({A, B} = {z, 𝐷} ↔ ((A = z B = 𝐷) (A = 𝐷 B = z)))) ↔ (𝐷 𝑌 → ({A, B} = {𝐶, 𝐷} ↔ ((A = 𝐶 B = 𝐷) (A = 𝐷 B = 𝐶))))))
28 preq2 3418 . . . . . . 7 (w = 𝐷 → {z, w} = {z, 𝐷})
2928eqeq2d 2029 . . . . . 6 (w = 𝐷 → ({x, y} = {z, w} ↔ {x, y} = {z, 𝐷}))
30 eqeq2 2027 . . . . . . . 8 (w = 𝐷 → (y = wy = 𝐷))
3130anbi2d 440 . . . . . . 7 (w = 𝐷 → ((x = z y = w) ↔ (x = z y = 𝐷)))
32 eqeq2 2027 . . . . . . . 8 (w = 𝐷 → (x = wx = 𝐷))
3332anbi1d 441 . . . . . . 7 (w = 𝐷 → ((x = w y = z) ↔ (x = 𝐷 y = z)))
3431, 33orbi12d 694 . . . . . 6 (w = 𝐷 → (((x = z y = w) (x = w y = z)) ↔ ((x = z y = 𝐷) (x = 𝐷 y = z))))
35 vex 2534 . . . . . . 7 x V
36 vex 2534 . . . . . . 7 y V
37 vex 2534 . . . . . . 7 z V
38 vex 2534 . . . . . . 7 w V
3935, 36, 37, 38preq12b 3511 . . . . . 6 ({x, y} = {z, w} ↔ ((x = z y = w) (x = w y = z)))
4029, 34, 39vtoclbg 2587 . . . . 5 (𝐷 𝑌 → ({x, y} = {z, 𝐷} ↔ ((x = z y = 𝐷) (x = 𝐷 y = z))))
4140a1i 9 . . . 4 ((x 𝑉 y 𝑊 z 𝑋) → (𝐷 𝑌 → ({x, y} = {z, 𝐷} ↔ ((x = z y = 𝐷) (x = 𝐷 y = z)))))
429, 18, 27, 41vtocl3ga 2596 . . 3 ((A 𝑉 B 𝑊 𝐶 𝑋) → (𝐷 𝑌 → ({A, B} = {𝐶, 𝐷} ↔ ((A = 𝐶 B = 𝐷) (A = 𝐷 B = 𝐶)))))
43423expa 1088 . 2 (((A 𝑉 B 𝑊) 𝐶 𝑋) → (𝐷 𝑌 → ({A, B} = {𝐶, 𝐷} ↔ ((A = 𝐶 B = 𝐷) (A = 𝐷 B = 𝐶)))))
4443impr 361 1 (((A 𝑉 B 𝑊) (𝐶 𝑋 𝐷 𝑌)) → ({A, B} = {𝐶, 𝐷} ↔ ((A = 𝐶 B = 𝐷) (A = 𝐷 B = 𝐶))))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   wo 616   w3a 871   = wceq 1226   wcel 1370  {cpr 3347
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-v 2533  df-un 2895  df-sn 3352  df-pr 3353
This theorem is referenced by:  prneimg  3515
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