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Theorem preq12b 3785
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 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)))

Proof of Theorem preq12b
StepHypRef Expression
1 preq12b.1 . . . . . 6 𝐴 ∈ V
21prid1 3713 . . . . 5 𝐴 ∈ {𝐴, 𝐵}
3 eleq2 2253 . . . . 5 ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 ∈ {𝐴, 𝐵} ↔ 𝐴 ∈ {𝐶, 𝐷}))
42, 3mpbii 148 . . . 4 ({𝐴, 𝐵} = {𝐶, 𝐷} → 𝐴 ∈ {𝐶, 𝐷})
51elpr 3628 . . . 4 (𝐴 ∈ {𝐶, 𝐷} ↔ (𝐴 = 𝐶𝐴 = 𝐷))
64, 5sylib 122 . . 3 ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 = 𝐶𝐴 = 𝐷))
7 preq1 3684 . . . . . . . 8 (𝐴 = 𝐶 → {𝐴, 𝐵} = {𝐶, 𝐵})
87eqeq1d 2198 . . . . . . 7 (𝐴 = 𝐶 → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ {𝐶, 𝐵} = {𝐶, 𝐷}))
9 preq12b.2 . . . . . . . 8 𝐵 ∈ V
10 preq12b.4 . . . . . . . 8 𝐷 ∈ V
119, 10preqr2 3784 . . . . . . 7 ({𝐶, 𝐵} = {𝐶, 𝐷} → 𝐵 = 𝐷)
128, 11biimtrdi 163 . . . . . 6 (𝐴 = 𝐶 → ({𝐴, 𝐵} = {𝐶, 𝐷} → 𝐵 = 𝐷))
1312com12 30 . . . . 5 ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 = 𝐶𝐵 = 𝐷))
1413ancld 325 . . . 4 ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 = 𝐶 → (𝐴 = 𝐶𝐵 = 𝐷)))
15 prcom 3683 . . . . . . 7 {𝐶, 𝐷} = {𝐷, 𝐶}
1615eqeq2i 2200 . . . . . 6 ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ {𝐴, 𝐵} = {𝐷, 𝐶})
17 preq1 3684 . . . . . . . . 9 (𝐴 = 𝐷 → {𝐴, 𝐵} = {𝐷, 𝐵})
1817eqeq1d 2198 . . . . . . . 8 (𝐴 = 𝐷 → ({𝐴, 𝐵} = {𝐷, 𝐶} ↔ {𝐷, 𝐵} = {𝐷, 𝐶}))
19 preq12b.3 . . . . . . . . 9 𝐶 ∈ V
209, 19preqr2 3784 . . . . . . . 8 ({𝐷, 𝐵} = {𝐷, 𝐶} → 𝐵 = 𝐶)
2118, 20biimtrdi 163 . . . . . . 7 (𝐴 = 𝐷 → ({𝐴, 𝐵} = {𝐷, 𝐶} → 𝐵 = 𝐶))
2221com12 30 . . . . . 6 ({𝐴, 𝐵} = {𝐷, 𝐶} → (𝐴 = 𝐷𝐵 = 𝐶))
2316, 22sylbi 121 . . . . 5 ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 = 𝐷𝐵 = 𝐶))
2423ancld 325 . . . 4 ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 = 𝐷 → (𝐴 = 𝐷𝐵 = 𝐶)))
2514, 24orim12d 787 . . 3 ({𝐴, 𝐵} = {𝐶, 𝐷} → ((𝐴 = 𝐶𝐴 = 𝐷) → ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
266, 25mpd 13 . 2 ({𝐴, 𝐵} = {𝐶, 𝐷} → ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)))
27 preq12 3686 . . 3 ((𝐴 = 𝐶𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷})
28 prcom 3683 . . . . 5 {𝐷, 𝐵} = {𝐵, 𝐷}
2917, 28eqtrdi 2238 . . . 4 (𝐴 = 𝐷 → {𝐴, 𝐵} = {𝐵, 𝐷})
30 preq1 3684 . . . 4 (𝐵 = 𝐶 → {𝐵, 𝐷} = {𝐶, 𝐷})
3129, 30sylan9eq 2242 . . 3 ((𝐴 = 𝐷𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐶, 𝐷})
3227, 31jaoi 717 . 2 (((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)) → {𝐴, 𝐵} = {𝐶, 𝐷})
3326, 32impbii 126 1 ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wo 709   = wceq 1364  wcel 2160  Vcvv 2752  {cpr 3608
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614
This theorem is referenced by:  prel12  3786  opthpr  3787  preq12bg  3788  preqsn  3790  opeqpr  4271  preleq  4572
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