Theorem preq12b 3750

Description: Equality relationship for two unordered pairs. (Contributed by NM, 17-Oct-1996.)

Hypotheses

Ref	Expression
preq12b.1	⊢ 𝐴 ∈ V
preq12b.2	⊢ 𝐵 ∈ V
preq12b.3	⊢ 𝐶 ∈ V
preq12b.4	⊢ 𝐷 ∈ V

Assertion

Ref	Expression
preq12b	⊢ ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶)))

Proof of Theorem preq12b

Step	Hyp	Ref	Expression
1		preq12b.1	. . . . . 6 ⊢ 𝐴 ∈ V
2	1	prid1 3682	. . . . 5 ⊢ 𝐴 ∈ {𝐴, 𝐵}
3		eleq2 2230	. . . . 5 ⊢ ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 ∈ {𝐴, 𝐵} ↔ 𝐴 ∈ {𝐶, 𝐷}))
4	2, 3	mpbii 147	. . . 4 ⊢ ({𝐴, 𝐵} = {𝐶, 𝐷} → 𝐴 ∈ {𝐶, 𝐷})
5	1	elpr 3597	. . . 4 ⊢ (𝐴 ∈ {𝐶, 𝐷} ↔ (𝐴 = 𝐶 ∨ 𝐴 = 𝐷))
6	4, 5	sylib 121	. . 3 ⊢ ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 = 𝐶 ∨ 𝐴 = 𝐷))
7		preq1 3653	. . . . . . . 8 ⊢ (𝐴 = 𝐶 → {𝐴, 𝐵} = {𝐶, 𝐵})
8	7	eqeq1d 2174	. . . . . . 7 ⊢ (𝐴 = 𝐶 → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ {𝐶, 𝐵} = {𝐶, 𝐷}))
9		preq12b.2	. . . . . . . 8 ⊢ 𝐵 ∈ V
10		preq12b.4	. . . . . . . 8 ⊢ 𝐷 ∈ V
11	9, 10	preqr2 3749	. . . . . . 7 ⊢ ({𝐶, 𝐵} = {𝐶, 𝐷} → 𝐵 = 𝐷)
12	8, 11	syl6bi 162	. . . . . 6 ⊢ (𝐴 = 𝐶 → ({𝐴, 𝐵} = {𝐶, 𝐷} → 𝐵 = 𝐷))
13	12	com12 30	. . . . 5 ⊢ ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 = 𝐶 → 𝐵 = 𝐷))
14	13	ancld 323	. . . 4 ⊢ ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 = 𝐶 → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
15		prcom 3652	. . . . . . 7 ⊢ {𝐶, 𝐷} = {𝐷, 𝐶}
16	15	eqeq2i 2176	. . . . . 6 ⊢ ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ {𝐴, 𝐵} = {𝐷, 𝐶})
17		preq1 3653	. . . . . . . . 9 ⊢ (𝐴 = 𝐷 → {𝐴, 𝐵} = {𝐷, 𝐵})
18	17	eqeq1d 2174	. . . . . . . 8 ⊢ (𝐴 = 𝐷 → ({𝐴, 𝐵} = {𝐷, 𝐶} ↔ {𝐷, 𝐵} = {𝐷, 𝐶}))
19		preq12b.3	. . . . . . . . 9 ⊢ 𝐶 ∈ V
20	9, 19	preqr2 3749	. . . . . . . 8 ⊢ ({𝐷, 𝐵} = {𝐷, 𝐶} → 𝐵 = 𝐶)
21	18, 20	syl6bi 162	. . . . . . 7 ⊢ (𝐴 = 𝐷 → ({𝐴, 𝐵} = {𝐷, 𝐶} → 𝐵 = 𝐶))
22	21	com12 30	. . . . . 6 ⊢ ({𝐴, 𝐵} = {𝐷, 𝐶} → (𝐴 = 𝐷 → 𝐵 = 𝐶))
23	16, 22	sylbi 120	. . . . 5 ⊢ ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 = 𝐷 → 𝐵 = 𝐶))
24	23	ancld 323	. . . 4 ⊢ ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 = 𝐷 → (𝐴 = 𝐷 ∧ 𝐵 = 𝐶)))
25	14, 24	orim12d 776	. . 3 ⊢ ({𝐴, 𝐵} = {𝐶, 𝐷} → ((𝐴 = 𝐶 ∨ 𝐴 = 𝐷) → ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶))))
26	6, 25	mpd 13	. 2 ⊢ ({𝐴, 𝐵} = {𝐶, 𝐷} → ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶)))
27		preq12 3655	. . 3 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷})
28		prcom 3652	. . . . 5 ⊢ {𝐷, 𝐵} = {𝐵, 𝐷}
29	17, 28	eqtrdi 2215	. . . 4 ⊢ (𝐴 = 𝐷 → {𝐴, 𝐵} = {𝐵, 𝐷})
30		preq1 3653	. . . 4 ⊢ (𝐵 = 𝐶 → {𝐵, 𝐷} = {𝐶, 𝐷})
31	29, 30	sylan9eq 2219	. . 3 ⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐶, 𝐷})
32	27, 31	jaoi 706	. 2 ⊢ (((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶)) → {𝐴, 𝐵} = {𝐶, 𝐷})
33	26, 32	impbii 125	1 ⊢ ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 698   = wceq 1343   ∈ wcel 2136  Vcvv 2726  {cpr 3577

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-sn 3582  df-pr 3583

This theorem is referenced by:  prel12  3751  opthpr  3752  preq12bg  3753  preqsn  3755  opeqpr  4231  preleq  4532