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Theorem prel12 3734
Description: Equality of 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
prel12 𝐴 = 𝐵 → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 ∈ {𝐶, 𝐷} ∧ 𝐵 ∈ {𝐶, 𝐷})))

Proof of Theorem prel12
StepHypRef Expression
1 preq12b.1 . . . . 5 𝐴 ∈ V
21prid1 3665 . . . 4 𝐴 ∈ {𝐴, 𝐵}
3 eleq2 2221 . . . 4 ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 ∈ {𝐴, 𝐵} ↔ 𝐴 ∈ {𝐶, 𝐷}))
42, 3mpbii 147 . . 3 ({𝐴, 𝐵} = {𝐶, 𝐷} → 𝐴 ∈ {𝐶, 𝐷})
5 preq12b.2 . . . . 5 𝐵 ∈ V
65prid2 3666 . . . 4 𝐵 ∈ {𝐴, 𝐵}
7 eleq2 2221 . . . 4 ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐵 ∈ {𝐴, 𝐵} ↔ 𝐵 ∈ {𝐶, 𝐷}))
86, 7mpbii 147 . . 3 ({𝐴, 𝐵} = {𝐶, 𝐷} → 𝐵 ∈ {𝐶, 𝐷})
94, 8jca 304 . 2 ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 ∈ {𝐶, 𝐷} ∧ 𝐵 ∈ {𝐶, 𝐷}))
101elpr 3581 . . . 4 (𝐴 ∈ {𝐶, 𝐷} ↔ (𝐴 = 𝐶𝐴 = 𝐷))
11 eqeq2 2167 . . . . . . . . . . . 12 (𝐵 = 𝐷 → (𝐴 = 𝐵𝐴 = 𝐷))
1211notbid 657 . . . . . . . . . . 11 (𝐵 = 𝐷 → (¬ 𝐴 = 𝐵 ↔ ¬ 𝐴 = 𝐷))
13 orel2 716 . . . . . . . . . . 11 𝐴 = 𝐷 → ((𝐴 = 𝐶𝐴 = 𝐷) → 𝐴 = 𝐶))
1412, 13syl6bi 162 . . . . . . . . . 10 (𝐵 = 𝐷 → (¬ 𝐴 = 𝐵 → ((𝐴 = 𝐶𝐴 = 𝐷) → 𝐴 = 𝐶)))
1514com3l 81 . . . . . . . . 9 𝐴 = 𝐵 → ((𝐴 = 𝐶𝐴 = 𝐷) → (𝐵 = 𝐷𝐴 = 𝐶)))
1615imp 123 . . . . . . . 8 ((¬ 𝐴 = 𝐵 ∧ (𝐴 = 𝐶𝐴 = 𝐷)) → (𝐵 = 𝐷𝐴 = 𝐶))
1716ancrd 324 . . . . . . 7 ((¬ 𝐴 = 𝐵 ∧ (𝐴 = 𝐶𝐴 = 𝐷)) → (𝐵 = 𝐷 → (𝐴 = 𝐶𝐵 = 𝐷)))
18 eqeq2 2167 . . . . . . . . . . . 12 (𝐵 = 𝐶 → (𝐴 = 𝐵𝐴 = 𝐶))
1918notbid 657 . . . . . . . . . . 11 (𝐵 = 𝐶 → (¬ 𝐴 = 𝐵 ↔ ¬ 𝐴 = 𝐶))
20 orel1 715 . . . . . . . . . . 11 𝐴 = 𝐶 → ((𝐴 = 𝐶𝐴 = 𝐷) → 𝐴 = 𝐷))
2119, 20syl6bi 162 . . . . . . . . . 10 (𝐵 = 𝐶 → (¬ 𝐴 = 𝐵 → ((𝐴 = 𝐶𝐴 = 𝐷) → 𝐴 = 𝐷)))
2221com3l 81 . . . . . . . . 9 𝐴 = 𝐵 → ((𝐴 = 𝐶𝐴 = 𝐷) → (𝐵 = 𝐶𝐴 = 𝐷)))
2322imp 123 . . . . . . . 8 ((¬ 𝐴 = 𝐵 ∧ (𝐴 = 𝐶𝐴 = 𝐷)) → (𝐵 = 𝐶𝐴 = 𝐷))
2423ancrd 324 . . . . . . 7 ((¬ 𝐴 = 𝐵 ∧ (𝐴 = 𝐶𝐴 = 𝐷)) → (𝐵 = 𝐶 → (𝐴 = 𝐷𝐵 = 𝐶)))
2517, 24orim12d 776 . . . . . 6 ((¬ 𝐴 = 𝐵 ∧ (𝐴 = 𝐶𝐴 = 𝐷)) → ((𝐵 = 𝐷𝐵 = 𝐶) → ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
265elpr 3581 . . . . . . 7 (𝐵 ∈ {𝐶, 𝐷} ↔ (𝐵 = 𝐶𝐵 = 𝐷))
27 orcom 718 . . . . . . 7 ((𝐵 = 𝐶𝐵 = 𝐷) ↔ (𝐵 = 𝐷𝐵 = 𝐶))
2826, 27bitri 183 . . . . . 6 (𝐵 ∈ {𝐶, 𝐷} ↔ (𝐵 = 𝐷𝐵 = 𝐶))
29 preq12b.3 . . . . . . 7 𝐶 ∈ V
30 preq12b.4 . . . . . . 7 𝐷 ∈ V
311, 5, 29, 30preq12b 3733 . . . . . 6 ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)))
3225, 28, 313imtr4g 204 . . . . 5 ((¬ 𝐴 = 𝐵 ∧ (𝐴 = 𝐶𝐴 = 𝐷)) → (𝐵 ∈ {𝐶, 𝐷} → {𝐴, 𝐵} = {𝐶, 𝐷}))
3332ex 114 . . . 4 𝐴 = 𝐵 → ((𝐴 = 𝐶𝐴 = 𝐷) → (𝐵 ∈ {𝐶, 𝐷} → {𝐴, 𝐵} = {𝐶, 𝐷})))
3410, 33syl5bi 151 . . 3 𝐴 = 𝐵 → (𝐴 ∈ {𝐶, 𝐷} → (𝐵 ∈ {𝐶, 𝐷} → {𝐴, 𝐵} = {𝐶, 𝐷})))
3534impd 252 . 2 𝐴 = 𝐵 → ((𝐴 ∈ {𝐶, 𝐷} ∧ 𝐵 ∈ {𝐶, 𝐷}) → {𝐴, 𝐵} = {𝐶, 𝐷}))
369, 35impbid2 142 1 𝐴 = 𝐵 → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 ∈ {𝐶, 𝐷} ∧ 𝐵 ∈ {𝐶, 𝐷})))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 698   = wceq 1335  wcel 2128  Vcvv 2712  {cpr 3561
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-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-sn 3566  df-pr 3567
This theorem is referenced by: (None)
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