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Theorem prel12 3516
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 3450 . . . 4 A {A, B}
3 eleq2 2083 . . . 4 ({A, B} = {𝐶, 𝐷} → (A {A, B} ↔ A {𝐶, 𝐷}))
42, 3mpbii 136 . . 3 ({A, B} = {𝐶, 𝐷} → A {𝐶, 𝐷})
5 preq12b.2 . . . . 5 B V
65prid2 3451 . . . 4 B {A, B}
7 eleq2 2083 . . . 4 ({A, B} = {𝐶, 𝐷} → (B {A, B} ↔ B {𝐶, 𝐷}))
86, 7mpbii 136 . . 3 ({A, B} = {𝐶, 𝐷} → B {𝐶, 𝐷})
94, 8jca 290 . 2 ({A, B} = {𝐶, 𝐷} → (A {𝐶, 𝐷} B {𝐶, 𝐷}))
101elpr 3368 . . . 4 (A {𝐶, 𝐷} ↔ (A = 𝐶 A = 𝐷))
11 eqeq2 2031 . . . . . . . . . . . 12 (B = 𝐷 → (A = BA = 𝐷))
1211notbid 579 . . . . . . . . . . 11 (B = 𝐷 → (¬ A = B ↔ ¬ A = 𝐷))
13 orel2 632 . . . . . . . . . . 11 A = 𝐷 → ((A = 𝐶 A = 𝐷) → A = 𝐶))
1412, 13syl6bi 152 . . . . . . . . . 10 (B = 𝐷 → (¬ A = B → ((A = 𝐶 A = 𝐷) → A = 𝐶)))
1514com3l 75 . . . . . . . . 9 A = B → ((A = 𝐶 A = 𝐷) → (B = 𝐷A = 𝐶)))
1615imp 115 . . . . . . . 8 ((¬ A = B (A = 𝐶 A = 𝐷)) → (B = 𝐷A = 𝐶))
1716ancrd 309 . . . . . . 7 ((¬ A = B (A = 𝐶 A = 𝐷)) → (B = 𝐷 → (A = 𝐶 B = 𝐷)))
18 eqeq2 2031 . . . . . . . . . . . 12 (B = 𝐶 → (A = BA = 𝐶))
1918notbid 579 . . . . . . . . . . 11 (B = 𝐶 → (¬ A = B ↔ ¬ A = 𝐶))
20 orel1 631 . . . . . . . . . . 11 A = 𝐶 → ((A = 𝐶 A = 𝐷) → A = 𝐷))
2119, 20syl6bi 152 . . . . . . . . . 10 (B = 𝐶 → (¬ A = B → ((A = 𝐶 A = 𝐷) → A = 𝐷)))
2221com3l 75 . . . . . . . . 9 A = B → ((A = 𝐶 A = 𝐷) → (B = 𝐶A = 𝐷)))
2322imp 115 . . . . . . . 8 ((¬ A = B (A = 𝐶 A = 𝐷)) → (B = 𝐶A = 𝐷))
2423ancrd 309 . . . . . . 7 ((¬ A = B (A = 𝐶 A = 𝐷)) → (B = 𝐶 → (A = 𝐷 B = 𝐶)))
2517, 24orim12d 687 . . . . . 6 ((¬ A = B (A = 𝐶 A = 𝐷)) → ((B = 𝐷 B = 𝐶) → ((A = 𝐶 B = 𝐷) (A = 𝐷 B = 𝐶))))
265elpr 3368 . . . . . . 7 (B {𝐶, 𝐷} ↔ (B = 𝐶 B = 𝐷))
27 orcom 634 . . . . . . 7 ((B = 𝐶 B = 𝐷) ↔ (B = 𝐷 B = 𝐶))
2826, 27bitri 173 . . . . . 6 (B {𝐶, 𝐷} ↔ (B = 𝐷 B = 𝐶))
29 preq12b.3 . . . . . . 7 𝐶 V
30 preq12b.4 . . . . . . 7 𝐷 V
311, 5, 29, 30preq12b 3515 . . . . . 6 ({A, B} = {𝐶, 𝐷} ↔ ((A = 𝐶 B = 𝐷) (A = 𝐷 B = 𝐶)))
3225, 28, 313imtr4g 194 . . . . 5 ((¬ A = B (A = 𝐶 A = 𝐷)) → (B {𝐶, 𝐷} → {A, B} = {𝐶, 𝐷}))
3332ex 108 . . . 4 A = B → ((A = 𝐶 A = 𝐷) → (B {𝐶, 𝐷} → {A, B} = {𝐶, 𝐷})))
3410, 33syl5bi 141 . . 3 A = B → (A {𝐶, 𝐷} → (B {𝐶, 𝐷} → {A, B} = {𝐶, 𝐷})))
3534impd 242 . 2 A = B → ((A {𝐶, 𝐷} B {𝐶, 𝐷}) → {A, B} = {𝐶, 𝐷}))
369, 35impbid2 131 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 97  wb 98   wo 616   = wceq 1228   wcel 1374  Vcvv 2535  {cpr 3351
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-un 2899  df-sn 3356  df-pr 3357
This theorem is referenced by: (None)
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