Theorem preq12bg 4808

Description: Closed form of preq12b 4805. (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 4689	. . . . . . 7 ⊢ (𝑥 = 𝐴 → {𝑥, 𝑦} = {𝐴, 𝑦})
2	1	eqeq1d 2737	. . . . . 6 ⊢ (𝑥 = 𝐴 → ({𝑥, 𝑦} = {𝑧, 𝐷} ↔ {𝐴, 𝑦} = {𝑧, 𝐷}))
3		eqeq1 2739	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝑧 ↔ 𝐴 = 𝑧))
4	3	anbi1d 632	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝑥 = 𝑧 ∧ 𝑦 = 𝐷) ↔ (𝐴 = 𝑧 ∧ 𝑦 = 𝐷)))
5		eqeq1 2739	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐷 ↔ 𝐴 = 𝐷))
6	5	anbi1d 632	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝑥 = 𝐷 ∧ 𝑦 = 𝑧) ↔ (𝐴 = 𝐷 ∧ 𝑦 = 𝑧)))
7	4, 6	orbi12d 919	. . . . . 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 4690	. . . . . . 7 ⊢ (𝑦 = 𝐵 → {𝐴, 𝑦} = {𝐴, 𝐵})
11	10	eqeq1d 2737	. . . . . 6 ⊢ (𝑦 = 𝐵 → ({𝐴, 𝑦} = {𝑧, 𝐷} ↔ {𝐴, 𝐵} = {𝑧, 𝐷}))
12		eqeq1 2739	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝑦 = 𝐷 ↔ 𝐵 = 𝐷))
13	12	anbi2d 631	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ((𝐴 = 𝑧 ∧ 𝑦 = 𝐷) ↔ (𝐴 = 𝑧 ∧ 𝐵 = 𝐷)))
14		eqeq1 2739	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝑦 = 𝑧 ↔ 𝐵 = 𝑧))
15	14	anbi2d 631	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ((𝐴 = 𝐷 ∧ 𝑦 = 𝑧) ↔ (𝐴 = 𝐷 ∧ 𝐵 = 𝑧)))
16	13, 15	orbi12d 919	. . . . . 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 4689	. . . . . . 7 ⊢ (𝑧 = 𝐶 → {𝑧, 𝐷} = {𝐶, 𝐷})
20	19	eqeq2d 2746	. . . . . 6 ⊢ (𝑧 = 𝐶 → ({𝐴, 𝐵} = {𝑧, 𝐷} ↔ {𝐴, 𝐵} = {𝐶, 𝐷}))
21		eqeq2 2747	. . . . . . . 8 ⊢ (𝑧 = 𝐶 → (𝐴 = 𝑧 ↔ 𝐴 = 𝐶))
22	21	anbi1d 632	. . . . . . 7 ⊢ (𝑧 = 𝐶 → ((𝐴 = 𝑧 ∧ 𝐵 = 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
23		eqeq2 2747	. . . . . . . 8 ⊢ (𝑧 = 𝐶 → (𝐵 = 𝑧 ↔ 𝐵 = 𝐶))
24	23	anbi2d 631	. . . . . . 7 ⊢ (𝑧 = 𝐶 → ((𝐴 = 𝐷 ∧ 𝐵 = 𝑧) ↔ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶)))
25	22, 24	orbi12d 919	. . . . . 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 4690	. . . . . . 7 ⊢ (𝑤 = 𝐷 → {𝑧, 𝑤} = {𝑧, 𝐷})
29	28	eqeq2d 2746	. . . . . 6 ⊢ (𝑤 = 𝐷 → ({𝑥, 𝑦} = {𝑧, 𝑤} ↔ {𝑥, 𝑦} = {𝑧, 𝐷}))
30		eqeq2 2747	. . . . . . . 8 ⊢ (𝑤 = 𝐷 → (𝑦 = 𝑤 ↔ 𝑦 = 𝐷))
31	30	anbi2d 631	. . . . . . 7 ⊢ (𝑤 = 𝐷 → ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝐷)))
32		eqeq2 2747	. . . . . . . 8 ⊢ (𝑤 = 𝐷 → (𝑥 = 𝑤 ↔ 𝑥 = 𝐷))
33	32	anbi1d 632	. . . . . . 7 ⊢ (𝑤 = 𝐷 → ((𝑥 = 𝑤 ∧ 𝑦 = 𝑧) ↔ (𝑥 = 𝐷 ∧ 𝑦 = 𝑧)))
34	31, 33	orbi12d 919	. . . . . 6 ⊢ (𝑤 = 𝐷 → (((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ∨ (𝑥 = 𝑤 ∧ 𝑦 = 𝑧)) ↔ ((𝑥 = 𝑧 ∧ 𝑦 = 𝐷) ∨ (𝑥 = 𝐷 ∧ 𝑦 = 𝑧))))
35		vex 3443	. . . . . . 7 ⊢ 𝑥 ∈ V
36		vex 3443	. . . . . . 7 ⊢ 𝑦 ∈ V
37		vex 3443	. . . . . . 7 ⊢ 𝑧 ∈ V
38		vex 3443	. . . . . . 7 ⊢ 𝑤 ∈ V
39	35, 36, 37, 38	preq12b 4805	. . . . . 6 ⊢ ({𝑥, 𝑦} = {𝑧, 𝑤} ↔ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ∨ (𝑥 = 𝑤 ∧ 𝑦 = 𝑧)))
40	29, 34, 39	vtoclbg 3513	. . . . 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 3537	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐷 ∈ 𝑌 → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶)))))
43	42	3expa 1119	. 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 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  {cpr 4581

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-v 3441  df-un 3905  df-sn 4580  df-pr 4582

This theorem is referenced by:  prneimg  4809  prneimg2  4810  pr1eqbg  4812  preqsnd  4814  preq12nebg  4818  opthprneg  4820  preleqALT  9528  pythagtriplem2  16747  pythagtrip  16764  upgrpredgv  29193  uhgr2edg  29262  usgredg2v  29281  2pthon3v  29997  opprb  47314  or2expropbi  47317  ich2exprop  47754  prsprel  47770  paireqne  47794  poprelb  47807  gpgvtxedg0  48346  gpgvtxedg1  48347