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Theorem prel12 3771
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 3698 . . . 4 𝐴 ∈ {𝐴, 𝐵}
3 eleq2 2241 . . . 4 ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 ∈ {𝐴, 𝐵} ↔ 𝐴 ∈ {𝐶, 𝐷}))
42, 3mpbii 148 . . 3 ({𝐴, 𝐵} = {𝐶, 𝐷} → 𝐴 ∈ {𝐶, 𝐷})
5 preq12b.2 . . . . 5 𝐵 ∈ V
65prid2 3699 . . . 4 𝐵 ∈ {𝐴, 𝐵}
7 eleq2 2241 . . . 4 ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐵 ∈ {𝐴, 𝐵} ↔ 𝐵 ∈ {𝐶, 𝐷}))
86, 7mpbii 148 . . 3 ({𝐴, 𝐵} = {𝐶, 𝐷} → 𝐵 ∈ {𝐶, 𝐷})
94, 8jca 306 . 2 ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 ∈ {𝐶, 𝐷} ∧ 𝐵 ∈ {𝐶, 𝐷}))
101elpr 3613 . . . 4 (𝐴 ∈ {𝐶, 𝐷} ↔ (𝐴 = 𝐶𝐴 = 𝐷))
11 eqeq2 2187 . . . . . . . . . . . 12 (𝐵 = 𝐷 → (𝐴 = 𝐵𝐴 = 𝐷))
1211notbid 667 . . . . . . . . . . 11 (𝐵 = 𝐷 → (¬ 𝐴 = 𝐵 ↔ ¬ 𝐴 = 𝐷))
13 orel2 726 . . . . . . . . . . 11 𝐴 = 𝐷 → ((𝐴 = 𝐶𝐴 = 𝐷) → 𝐴 = 𝐶))
1412, 13syl6bi 163 . . . . . . . . . 10 (𝐵 = 𝐷 → (¬ 𝐴 = 𝐵 → ((𝐴 = 𝐶𝐴 = 𝐷) → 𝐴 = 𝐶)))
1514com3l 81 . . . . . . . . 9 𝐴 = 𝐵 → ((𝐴 = 𝐶𝐴 = 𝐷) → (𝐵 = 𝐷𝐴 = 𝐶)))
1615imp 124 . . . . . . . 8 ((¬ 𝐴 = 𝐵 ∧ (𝐴 = 𝐶𝐴 = 𝐷)) → (𝐵 = 𝐷𝐴 = 𝐶))
1716ancrd 326 . . . . . . 7 ((¬ 𝐴 = 𝐵 ∧ (𝐴 = 𝐶𝐴 = 𝐷)) → (𝐵 = 𝐷 → (𝐴 = 𝐶𝐵 = 𝐷)))
18 eqeq2 2187 . . . . . . . . . . . 12 (𝐵 = 𝐶 → (𝐴 = 𝐵𝐴 = 𝐶))
1918notbid 667 . . . . . . . . . . 11 (𝐵 = 𝐶 → (¬ 𝐴 = 𝐵 ↔ ¬ 𝐴 = 𝐶))
20 orel1 725 . . . . . . . . . . 11 𝐴 = 𝐶 → ((𝐴 = 𝐶𝐴 = 𝐷) → 𝐴 = 𝐷))
2119, 20syl6bi 163 . . . . . . . . . 10 (𝐵 = 𝐶 → (¬ 𝐴 = 𝐵 → ((𝐴 = 𝐶𝐴 = 𝐷) → 𝐴 = 𝐷)))
2221com3l 81 . . . . . . . . 9 𝐴 = 𝐵 → ((𝐴 = 𝐶𝐴 = 𝐷) → (𝐵 = 𝐶𝐴 = 𝐷)))
2322imp 124 . . . . . . . 8 ((¬ 𝐴 = 𝐵 ∧ (𝐴 = 𝐶𝐴 = 𝐷)) → (𝐵 = 𝐶𝐴 = 𝐷))
2423ancrd 326 . . . . . . 7 ((¬ 𝐴 = 𝐵 ∧ (𝐴 = 𝐶𝐴 = 𝐷)) → (𝐵 = 𝐶 → (𝐴 = 𝐷𝐵 = 𝐶)))
2517, 24orim12d 786 . . . . . 6 ((¬ 𝐴 = 𝐵 ∧ (𝐴 = 𝐶𝐴 = 𝐷)) → ((𝐵 = 𝐷𝐵 = 𝐶) → ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
265elpr 3613 . . . . . . 7 (𝐵 ∈ {𝐶, 𝐷} ↔ (𝐵 = 𝐶𝐵 = 𝐷))
27 orcom 728 . . . . . . 7 ((𝐵 = 𝐶𝐵 = 𝐷) ↔ (𝐵 = 𝐷𝐵 = 𝐶))
2826, 27bitri 184 . . . . . 6 (𝐵 ∈ {𝐶, 𝐷} ↔ (𝐵 = 𝐷𝐵 = 𝐶))
29 preq12b.3 . . . . . . 7 𝐶 ∈ V
30 preq12b.4 . . . . . . 7 𝐷 ∈ V
311, 5, 29, 30preq12b 3770 . . . . . 6 ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)))
3225, 28, 313imtr4g 205 . . . . 5 ((¬ 𝐴 = 𝐵 ∧ (𝐴 = 𝐶𝐴 = 𝐷)) → (𝐵 ∈ {𝐶, 𝐷} → {𝐴, 𝐵} = {𝐶, 𝐷}))
3332ex 115 . . . 4 𝐴 = 𝐵 → ((𝐴 = 𝐶𝐴 = 𝐷) → (𝐵 ∈ {𝐶, 𝐷} → {𝐴, 𝐵} = {𝐶, 𝐷})))
3410, 33biimtrid 152 . . 3 𝐴 = 𝐵 → (𝐴 ∈ {𝐶, 𝐷} → (𝐵 ∈ {𝐶, 𝐷} → {𝐴, 𝐵} = {𝐶, 𝐷})))
3534impd 254 . 2 𝐴 = 𝐵 → ((𝐴 ∈ {𝐶, 𝐷} ∧ 𝐵 ∈ {𝐶, 𝐷}) → {𝐴, 𝐵} = {𝐶, 𝐷}))
369, 35impbid2 143 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 104  wb 105  wo 708   = wceq 1353  wcel 2148  Vcvv 2737  {cpr 3593
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-in1 614  ax-in2 615  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-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 3598  df-pr 3599
This theorem is referenced by: (None)
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