Theorem preq12bg 3771

Description: Closed form of preq12b 3768. (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 3668	. . . . . . 7 ⊢ (𝑥 = 𝐴 → {𝑥, 𝑦} = {𝐴, 𝑦})
2	1	eqeq1d 2186	. . . . . 6 ⊢ (𝑥 = 𝐴 → ({𝑥, 𝑦} = {𝑧, 𝐷} ↔ {𝐴, 𝑦} = {𝑧, 𝐷}))
3		eqeq1 2184	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝑧 ↔ 𝐴 = 𝑧))
4	3	anbi1d 465	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝑥 = 𝑧 ∧ 𝑦 = 𝐷) ↔ (𝐴 = 𝑧 ∧ 𝑦 = 𝐷)))
5		eqeq1 2184	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐷 ↔ 𝐴 = 𝐷))
6	5	anbi1d 465	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝑥 = 𝐷 ∧ 𝑦 = 𝑧) ↔ (𝐴 = 𝐷 ∧ 𝑦 = 𝑧)))
7	4, 6	orbi12d 793	. . . . . 6 ⊢ (𝑥 = 𝐴 → (((𝑥 = 𝑧 ∧ 𝑦 = 𝐷) ∨ (𝑥 = 𝐷 ∧ 𝑦 = 𝑧)) ↔ ((𝐴 = 𝑧 ∧ 𝑦 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝑦 = 𝑧))))
8	2, 7	bibi12d 235	. . . . 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 230	. . . 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 3669	. . . . . . 7 ⊢ (𝑦 = 𝐵 → {𝐴, 𝑦} = {𝐴, 𝐵})
11	10	eqeq1d 2186	. . . . . 6 ⊢ (𝑦 = 𝐵 → ({𝐴, 𝑦} = {𝑧, 𝐷} ↔ {𝐴, 𝐵} = {𝑧, 𝐷}))
12		eqeq1 2184	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝑦 = 𝐷 ↔ 𝐵 = 𝐷))
13	12	anbi2d 464	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ((𝐴 = 𝑧 ∧ 𝑦 = 𝐷) ↔ (𝐴 = 𝑧 ∧ 𝐵 = 𝐷)))
14		eqeq1 2184	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝑦 = 𝑧 ↔ 𝐵 = 𝑧))
15	14	anbi2d 464	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ((𝐴 = 𝐷 ∧ 𝑦 = 𝑧) ↔ (𝐴 = 𝐷 ∧ 𝐵 = 𝑧)))
16	13, 15	orbi12d 793	. . . . . 6 ⊢ (𝑦 = 𝐵 → (((𝐴 = 𝑧 ∧ 𝑦 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝑦 = 𝑧)) ↔ ((𝐴 = 𝑧 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝑧))))
17	11, 16	bibi12d 235	. . . . 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 230	. . . 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 3668	. . . . . . 7 ⊢ (𝑧 = 𝐶 → {𝑧, 𝐷} = {𝐶, 𝐷})
20	19	eqeq2d 2189	. . . . . 6 ⊢ (𝑧 = 𝐶 → ({𝐴, 𝐵} = {𝑧, 𝐷} ↔ {𝐴, 𝐵} = {𝐶, 𝐷}))
21		eqeq2 2187	. . . . . . . 8 ⊢ (𝑧 = 𝐶 → (𝐴 = 𝑧 ↔ 𝐴 = 𝐶))
22	21	anbi1d 465	. . . . . . 7 ⊢ (𝑧 = 𝐶 → ((𝐴 = 𝑧 ∧ 𝐵 = 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
23		eqeq2 2187	. . . . . . . 8 ⊢ (𝑧 = 𝐶 → (𝐵 = 𝑧 ↔ 𝐵 = 𝐶))
24	23	anbi2d 464	. . . . . . 7 ⊢ (𝑧 = 𝐶 → ((𝐴 = 𝐷 ∧ 𝐵 = 𝑧) ↔ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶)))
25	22, 24	orbi12d 793	. . . . . 6 ⊢ (𝑧 = 𝐶 → (((𝐴 = 𝑧 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝑧)) ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶))))
26	20, 25	bibi12d 235	. . . . 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 230	. . . 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 3669	. . . . . . 7 ⊢ (𝑤 = 𝐷 → {𝑧, 𝑤} = {𝑧, 𝐷})
29	28	eqeq2d 2189	. . . . . 6 ⊢ (𝑤 = 𝐷 → ({𝑥, 𝑦} = {𝑧, 𝑤} ↔ {𝑥, 𝑦} = {𝑧, 𝐷}))
30		eqeq2 2187	. . . . . . . 8 ⊢ (𝑤 = 𝐷 → (𝑦 = 𝑤 ↔ 𝑦 = 𝐷))
31	30	anbi2d 464	. . . . . . 7 ⊢ (𝑤 = 𝐷 → ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝐷)))
32		eqeq2 2187	. . . . . . . 8 ⊢ (𝑤 = 𝐷 → (𝑥 = 𝑤 ↔ 𝑥 = 𝐷))
33	32	anbi1d 465	. . . . . . 7 ⊢ (𝑤 = 𝐷 → ((𝑥 = 𝑤 ∧ 𝑦 = 𝑧) ↔ (𝑥 = 𝐷 ∧ 𝑦 = 𝑧)))
34	31, 33	orbi12d 793	. . . . . 6 ⊢ (𝑤 = 𝐷 → (((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ∨ (𝑥 = 𝑤 ∧ 𝑦 = 𝑧)) ↔ ((𝑥 = 𝑧 ∧ 𝑦 = 𝐷) ∨ (𝑥 = 𝐷 ∧ 𝑦 = 𝑧))))
35		vex 2740	. . . . . . 7 ⊢ 𝑥 ∈ V
36		vex 2740	. . . . . . 7 ⊢ 𝑦 ∈ V
37		vex 2740	. . . . . . 7 ⊢ 𝑧 ∈ V
38		vex 2740	. . . . . . 7 ⊢ 𝑤 ∈ V
39	35, 36, 37, 38	preq12b 3768	. . . . . 6 ⊢ ({𝑥, 𝑦} = {𝑧, 𝑤} ↔ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ∨ (𝑥 = 𝑤 ∧ 𝑦 = 𝑧)))
40	29, 34, 39	vtoclbg 2798	. . . . 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 2807	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐷 ∈ 𝑌 → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶)))))
43	42	3expa 1203	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐶 ∈ 𝑋) → (𝐷 ∈ 𝑌 → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶)))))
44	43	impr 379	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 104   ↔ wb 105   ∨ wo 708   ∧ w3a 978   = wceq 1353   ∈ wcel 2148  {cpr 3592

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-sn 3597  df-pr 3598

This theorem is referenced by:  prneimg  3772  pythagtriplem2  12249  pythagtrip  12266