Theorem preq12bg 4804

Description: Closed form of preq12b 4801. (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 4685	. . . . . . 7 ⊢ (𝑥 = 𝐴 → {𝑥, 𝑦} = {𝐴, 𝑦})
2	1	eqeq1d 2731	. . . . . 6 ⊢ (𝑥 = 𝐴 → ({𝑥, 𝑦} = {𝑧, 𝐷} ↔ {𝐴, 𝑦} = {𝑧, 𝐷}))
3		eqeq1 2733	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝑧 ↔ 𝐴 = 𝑧))
4	3	anbi1d 631	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝑥 = 𝑧 ∧ 𝑦 = 𝐷) ↔ (𝐴 = 𝑧 ∧ 𝑦 = 𝐷)))
5		eqeq1 2733	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐷 ↔ 𝐴 = 𝐷))
6	5	anbi1d 631	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝑥 = 𝐷 ∧ 𝑦 = 𝑧) ↔ (𝐴 = 𝐷 ∧ 𝑦 = 𝑧)))
7	4, 6	orbi12d 918	. . . . . 6 ⊢ (𝑥 = 𝐴 → (((𝑥 = 𝑧 ∧ 𝑦 = 𝐷) ∨ (𝑥 = 𝐷 ∧ 𝑦 = 𝑧)) ↔ ((𝐴 = 𝑧 ∧ 𝑦 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝑦 = 𝑧))))
8	2, 7	bibi12d 345	. . . . 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 340	. . . 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 4686	. . . . . . 7 ⊢ (𝑦 = 𝐵 → {𝐴, 𝑦} = {𝐴, 𝐵})
11	10	eqeq1d 2731	. . . . . 6 ⊢ (𝑦 = 𝐵 → ({𝐴, 𝑦} = {𝑧, 𝐷} ↔ {𝐴, 𝐵} = {𝑧, 𝐷}))
12		eqeq1 2733	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝑦 = 𝐷 ↔ 𝐵 = 𝐷))
13	12	anbi2d 630	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ((𝐴 = 𝑧 ∧ 𝑦 = 𝐷) ↔ (𝐴 = 𝑧 ∧ 𝐵 = 𝐷)))
14		eqeq1 2733	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝑦 = 𝑧 ↔ 𝐵 = 𝑧))
15	14	anbi2d 630	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ((𝐴 = 𝐷 ∧ 𝑦 = 𝑧) ↔ (𝐴 = 𝐷 ∧ 𝐵 = 𝑧)))
16	13, 15	orbi12d 918	. . . . . 6 ⊢ (𝑦 = 𝐵 → (((𝐴 = 𝑧 ∧ 𝑦 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝑦 = 𝑧)) ↔ ((𝐴 = 𝑧 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝑧))))
17	11, 16	bibi12d 345	. . . . 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 340	. . . 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 4685	. . . . . . 7 ⊢ (𝑧 = 𝐶 → {𝑧, 𝐷} = {𝐶, 𝐷})
20	19	eqeq2d 2740	. . . . . 6 ⊢ (𝑧 = 𝐶 → ({𝐴, 𝐵} = {𝑧, 𝐷} ↔ {𝐴, 𝐵} = {𝐶, 𝐷}))
21		eqeq2 2741	. . . . . . . 8 ⊢ (𝑧 = 𝐶 → (𝐴 = 𝑧 ↔ 𝐴 = 𝐶))
22	21	anbi1d 631	. . . . . . 7 ⊢ (𝑧 = 𝐶 → ((𝐴 = 𝑧 ∧ 𝐵 = 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
23		eqeq2 2741	. . . . . . . 8 ⊢ (𝑧 = 𝐶 → (𝐵 = 𝑧 ↔ 𝐵 = 𝐶))
24	23	anbi2d 630	. . . . . . 7 ⊢ (𝑧 = 𝐶 → ((𝐴 = 𝐷 ∧ 𝐵 = 𝑧) ↔ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶)))
25	22, 24	orbi12d 918	. . . . . 6 ⊢ (𝑧 = 𝐶 → (((𝐴 = 𝑧 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝑧)) ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶))))
26	20, 25	bibi12d 345	. . . . 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 340	. . . 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 4686	. . . . . . 7 ⊢ (𝑤 = 𝐷 → {𝑧, 𝑤} = {𝑧, 𝐷})
29	28	eqeq2d 2740	. . . . . 6 ⊢ (𝑤 = 𝐷 → ({𝑥, 𝑦} = {𝑧, 𝑤} ↔ {𝑥, 𝑦} = {𝑧, 𝐷}))
30		eqeq2 2741	. . . . . . . 8 ⊢ (𝑤 = 𝐷 → (𝑦 = 𝑤 ↔ 𝑦 = 𝐷))
31	30	anbi2d 630	. . . . . . 7 ⊢ (𝑤 = 𝐷 → ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝐷)))
32		eqeq2 2741	. . . . . . . 8 ⊢ (𝑤 = 𝐷 → (𝑥 = 𝑤 ↔ 𝑥 = 𝐷))
33	32	anbi1d 631	. . . . . . 7 ⊢ (𝑤 = 𝐷 → ((𝑥 = 𝑤 ∧ 𝑦 = 𝑧) ↔ (𝑥 = 𝐷 ∧ 𝑦 = 𝑧)))
34	31, 33	orbi12d 918	. . . . . 6 ⊢ (𝑤 = 𝐷 → (((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ∨ (𝑥 = 𝑤 ∧ 𝑦 = 𝑧)) ↔ ((𝑥 = 𝑧 ∧ 𝑦 = 𝐷) ∨ (𝑥 = 𝐷 ∧ 𝑦 = 𝑧))))
35		vex 3440	. . . . . . 7 ⊢ 𝑥 ∈ V
36		vex 3440	. . . . . . 7 ⊢ 𝑦 ∈ V
37		vex 3440	. . . . . . 7 ⊢ 𝑧 ∈ V
38		vex 3440	. . . . . . 7 ⊢ 𝑤 ∈ V
39	35, 36, 37, 38	preq12b 4801	. . . . . 6 ⊢ ({𝑥, 𝑦} = {𝑧, 𝑤} ↔ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ∨ (𝑥 = 𝑤 ∧ 𝑦 = 𝑧)))
40	29, 34, 39	vtoclbg 3512	. . . . 5 ⊢ (𝐷 ∈ 𝑌 → ({𝑥, 𝑦} = {𝑧, 𝐷} ↔ ((𝑥 = 𝑧 ∧ 𝑦 = 𝐷) ∨ (𝑥 = 𝐷 ∧ 𝑦 = 𝑧))))
41	40	a1i 11	. . . 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 3538	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐷 ∈ 𝑌 → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶)))))
43	42	3expa 1118	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐶 ∈ 𝑋) → (𝐷 ∈ 𝑌 → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶)))))
44	43	impr 454	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   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  {cpr 4579

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3438  df-un 3908  df-sn 4578  df-pr 4580

This theorem is referenced by:  prneimg  4805  prneimg2  4806  pr1eqbg  4808  preqsnd  4810  preq12nebg  4814  opthprneg  4816  preleqALT  9513  pythagtriplem2  16729  pythagtrip  16746  upgrpredgv  29088  uhgr2edg  29157  usgredg2v  29176  2pthon3v  29892  opprb  47035  or2expropbi  47038  ich2exprop  47475  prsprel  47491  paireqne  47515  poprelb  47528  gpgvtxedg0  48067  gpgvtxedg1  48068