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.

Step	Hyp	Ref	Expression
1		preq1 3417	. . . . . . 7 ⊢ (x = A → {x, y} = {A, y})
2	1	eqeq1d 2026	. . . . . 6 ⊢ (x = A → ({x, y} = {z, 𝐷} ↔ {A, y} = {z, 𝐷}))
3		eqeq1 2024	. . . . . . . 8 ⊢ (x = A → (x = z ↔ A = z))
4	3	anbi1d 441	. . . . . . 7 ⊢ (x = A → ((x = z ∧ y = 𝐷) ↔ (A = z ∧ y = 𝐷)))
5		eqeq1 2024	. . . . . . . 8 ⊢ (x = A → (x = 𝐷 ↔ A = 𝐷))
6	5	anbi1d 441	. . . . . . 7 ⊢ (x = A → ((x = 𝐷 ∧ y = z) ↔ (A = 𝐷 ∧ y = z)))
7	4, 6	orbi12d 694	. . . . . 6 ⊢ (x = A → (((x = z ∧ y = 𝐷) ∨ (x = 𝐷 ∧ y = z)) ↔ ((A = z ∧ y = 𝐷) ∨ (A = 𝐷 ∧ y = z))))
8	2, 7	bibi12d 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)))))
9	8	imbi2d 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})
11	10	eqeq1d 2026	. . . . . 6 ⊢ (y = B → ({A, y} = {z, 𝐷} ↔ {A, B} = {z, 𝐷}))
12		eqeq1 2024	. . . . . . . 8 ⊢ (y = B → (y = 𝐷 ↔ B = 𝐷))
13	12	anbi2d 440	. . . . . . 7 ⊢ (y = B → ((A = z ∧ y = 𝐷) ↔ (A = z ∧ B = 𝐷)))
14		eqeq1 2024	. . . . . . . 8 ⊢ (y = B → (y = z ↔ B = z))
15	14	anbi2d 440	. . . . . . 7 ⊢ (y = B → ((A = 𝐷 ∧ y = z) ↔ (A = 𝐷 ∧ B = z)))
16	13, 15	orbi12d 694	. . . . . 6 ⊢ (y = B → (((A = z ∧ y = 𝐷) ∨ (A = 𝐷 ∧ y = z)) ↔ ((A = z ∧ B = 𝐷) ∨ (A = 𝐷 ∧ B = z))))
17	11, 16	bibi12d 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)))))
18	17	imbi2d 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})
20	19	eqeq2d 2029	. . . . . 6 ⊢ (z = 𝐶 → ({A, B} = {z, 𝐷} ↔ {A, B} = {𝐶, 𝐷}))
21		eqeq2 2027	. . . . . . . 8 ⊢ (z = 𝐶 → (A = z ↔ A = 𝐶))
22	21	anbi1d 441	. . . . . . 7 ⊢ (z = 𝐶 → ((A = z ∧ B = 𝐷) ↔ (A = 𝐶 ∧ B = 𝐷)))
23		eqeq2 2027	. . . . . . . 8 ⊢ (z = 𝐶 → (B = z ↔ B = 𝐶))
24	23	anbi2d 440	. . . . . . 7 ⊢ (z = 𝐶 → ((A = 𝐷 ∧ B = z) ↔ (A = 𝐷 ∧ B = 𝐶)))
25	22, 24	orbi12d 694	. . . . . 6 ⊢ (z = 𝐶 → (((A = z ∧ B = 𝐷) ∨ (A = 𝐷 ∧ B = z)) ↔ ((A = 𝐶 ∧ B = 𝐷) ∨ (A = 𝐷 ∧ B = 𝐶))))
26	20, 25	bibi12d 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)))))
27	26	imbi2d 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})
29	28	eqeq2d 2029	. . . . . 6 ⊢ (w = 𝐷 → ({x, y} = {z, w} ↔ {x, y} = {z, 𝐷}))
30		eqeq2 2027	. . . . . . . 8 ⊢ (w = 𝐷 → (y = w ↔ y = 𝐷))
31	30	anbi2d 440	. . . . . . 7 ⊢ (w = 𝐷 → ((x = z ∧ y = w) ↔ (x = z ∧ y = 𝐷)))
32		eqeq2 2027	. . . . . . . 8 ⊢ (w = 𝐷 → (x = w ↔ x = 𝐷))
33	32	anbi1d 441	. . . . . . 7 ⊢ (w = 𝐷 → ((x = w ∧ y = z) ↔ (x = 𝐷 ∧ y = z)))
34	31, 33	orbi12d 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
39	35, 36, 37, 38	preq12b 3511	. . . . . 6 ⊢ ({x, y} = {z, w} ↔ ((x = z ∧ y = w) ∨ (x = w ∧ y = z)))
40	29, 34, 39	vtoclbg 2587	. . . . 5 ⊢ (𝐷 ∈ 𝑌 → ({x, y} = {z, 𝐷} ↔ ((x = z ∧ y = 𝐷) ∨ (x = 𝐷 ∧ y = z))))
41	40	a1i 9	. . . 4 ⊢ ((x ∈ 𝑉 ∧ y ∈ 𝑊 ∧ z ∈ 𝑋) → (𝐷 ∈ 𝑌 → ({x, y} = {z, 𝐷} ↔ ((x = z ∧ y = 𝐷) ∨ (x = 𝐷 ∧ y = z)))))
42	9, 18, 27, 41	vtocl3ga 2596	. . 3 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐷 ∈ 𝑌 → ({A, B} = {𝐶, 𝐷} ↔ ((A = 𝐶 ∧ B = 𝐷) ∨ (A = 𝐷 ∧ B = 𝐶)))))
43	42	3expa 1088	. 2 ⊢ (((A ∈ 𝑉 ∧ B ∈ 𝑊) ∧ 𝐶 ∈ 𝑋) → (𝐷 ∈ 𝑌 → ({A, B} = {𝐶, 𝐷} ↔ ((A = 𝐶 ∧ B = 𝐷) ∨ (A = 𝐷 ∧ B = 𝐶)))))
44	43	impr 361	1 ⊢ (((A ∈ 𝑉 ∧ B ∈ 𝑊) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌)) → ({A, B} = {𝐶, 𝐷} ↔ ((A = 𝐶 ∧ B = 𝐷) ∨ (A = 𝐷 ∧ B = 𝐶))))

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