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Theorem preq12b 4806
Description: Equality relationship for two unordered pairs. (Contributed by NM, 17-Oct-1996.)
Hypotheses
Ref Expression
preqr1.a 𝐴 ∈ V
preqr1.b 𝐵 ∈ V
preq12b.c 𝐶 ∈ V
preq12b.d 𝐷 ∈ V
Assertion
Ref Expression
preq12b ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)))

Proof of Theorem preq12b
StepHypRef Expression
1 preqr1.a . . . . . 6 𝐴 ∈ V
21prid1 4721 . . . . 5 𝐴 ∈ {𝐴, 𝐵}
3 eleq2 2826 . . . . 5 ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 ∈ {𝐴, 𝐵} ↔ 𝐴 ∈ {𝐶, 𝐷}))
42, 3mpbii 232 . . . 4 ({𝐴, 𝐵} = {𝐶, 𝐷} → 𝐴 ∈ {𝐶, 𝐷})
51elpr 4607 . . . 4 (𝐴 ∈ {𝐶, 𝐷} ↔ (𝐴 = 𝐶𝐴 = 𝐷))
64, 5sylib 217 . . 3 ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 = 𝐶𝐴 = 𝐷))
7 preq1 4692 . . . . . . . 8 (𝐴 = 𝐶 → {𝐴, 𝐵} = {𝐶, 𝐵})
87eqeq1d 2739 . . . . . . 7 (𝐴 = 𝐶 → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ {𝐶, 𝐵} = {𝐶, 𝐷}))
9 preqr1.b . . . . . . . 8 𝐵 ∈ V
10 preq12b.d . . . . . . . 8 𝐷 ∈ V
119, 10preqr2 4805 . . . . . . 7 ({𝐶, 𝐵} = {𝐶, 𝐷} → 𝐵 = 𝐷)
128, 11syl6bi 252 . . . . . 6 (𝐴 = 𝐶 → ({𝐴, 𝐵} = {𝐶, 𝐷} → 𝐵 = 𝐷))
1312com12 32 . . . . 5 ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 = 𝐶𝐵 = 𝐷))
1413ancld 551 . . . 4 ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 = 𝐶 → (𝐴 = 𝐶𝐵 = 𝐷)))
15 prcom 4691 . . . . . . 7 {𝐶, 𝐷} = {𝐷, 𝐶}
1615eqeq2i 2750 . . . . . 6 ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ {𝐴, 𝐵} = {𝐷, 𝐶})
17 preq1 4692 . . . . . . . . 9 (𝐴 = 𝐷 → {𝐴, 𝐵} = {𝐷, 𝐵})
1817eqeq1d 2739 . . . . . . . 8 (𝐴 = 𝐷 → ({𝐴, 𝐵} = {𝐷, 𝐶} ↔ {𝐷, 𝐵} = {𝐷, 𝐶}))
19 preq12b.c . . . . . . . . 9 𝐶 ∈ V
209, 19preqr2 4805 . . . . . . . 8 ({𝐷, 𝐵} = {𝐷, 𝐶} → 𝐵 = 𝐶)
2118, 20syl6bi 252 . . . . . . 7 (𝐴 = 𝐷 → ({𝐴, 𝐵} = {𝐷, 𝐶} → 𝐵 = 𝐶))
2221com12 32 . . . . . 6 ({𝐴, 𝐵} = {𝐷, 𝐶} → (𝐴 = 𝐷𝐵 = 𝐶))
2316, 22sylbi 216 . . . . 5 ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 = 𝐷𝐵 = 𝐶))
2423ancld 551 . . . 4 ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 = 𝐷 → (𝐴 = 𝐷𝐵 = 𝐶)))
2514, 24orim12d 963 . . 3 ({𝐴, 𝐵} = {𝐶, 𝐷} → ((𝐴 = 𝐶𝐴 = 𝐷) → ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
266, 25mpd 15 . 2 ({𝐴, 𝐵} = {𝐶, 𝐷} → ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)))
27 preq12 4694 . . 3 ((𝐴 = 𝐶𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷})
28 preq12 4694 . . . 4 ((𝐴 = 𝐷𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐷, 𝐶})
29 prcom 4691 . . . 4 {𝐷, 𝐶} = {𝐶, 𝐷}
3028, 29eqtrdi 2793 . . 3 ((𝐴 = 𝐷𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐶, 𝐷})
3127, 30jaoi 855 . 2 (((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)) → {𝐴, 𝐵} = {𝐶, 𝐷})
3226, 31impbii 208 1 ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wo 845   = wceq 1541  wcel 2106  Vcvv 3443  {cpr 4586
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-v 3445  df-un 3913  df-sn 4585  df-pr 4587
This theorem is referenced by:  opthpr  4807  preq12bg  4809  opeqpr  5460  opthhausdorff0  5473  axlowdimlem13  27731  upgrwlkdvdelem  28512  altopthsn  34477  sprsymrelfolem2  45579  prproropf1olem4  45592  reuopreuprim  45612  isomuspgrlem1  45913  rrx2xpref1o  46698
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