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Theorem preq12b 3532
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 = 𝐶 B = 𝐷) (A = 𝐷 B = 𝐶)))

Proof of Theorem preq12b
StepHypRef Expression
1 preq12b.1 . . . . . 6 A V
21prid1 3467 . . . . 5 A {A, B}
3 eleq2 2098 . . . . 5 ({A, B} = {𝐶, 𝐷} → (A {A, B} ↔ A {𝐶, 𝐷}))
42, 3mpbii 136 . . . 4 ({A, B} = {𝐶, 𝐷} → A {𝐶, 𝐷})
51elpr 3385 . . . 4 (A {𝐶, 𝐷} ↔ (A = 𝐶 A = 𝐷))
64, 5sylib 127 . . 3 ({A, B} = {𝐶, 𝐷} → (A = 𝐶 A = 𝐷))
7 preq1 3438 . . . . . . . 8 (A = 𝐶 → {A, B} = {𝐶, B})
87eqeq1d 2045 . . . . . . 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 3531 . . . . . . 7 ({𝐶, B} = {𝐶, 𝐷} → B = 𝐷)
128, 11syl6bi 152 . . . . . 6 (A = 𝐶 → ({A, B} = {𝐶, 𝐷} → B = 𝐷))
1312com12 27 . . . . 5 ({A, B} = {𝐶, 𝐷} → (A = 𝐶B = 𝐷))
1413ancld 308 . . . 4 ({A, B} = {𝐶, 𝐷} → (A = 𝐶 → (A = 𝐶 B = 𝐷)))
15 prcom 3437 . . . . . . 7 {𝐶, 𝐷} = {𝐷, 𝐶}
1615eqeq2i 2047 . . . . . 6 ({A, B} = {𝐶, 𝐷} ↔ {A, B} = {𝐷, 𝐶})
17 preq1 3438 . . . . . . . . 9 (A = 𝐷 → {A, B} = {𝐷, B})
1817eqeq1d 2045 . . . . . . . 8 (A = 𝐷 → ({A, B} = {𝐷, 𝐶} ↔ {𝐷, B} = {𝐷, 𝐶}))
19 preq12b.3 . . . . . . . . 9 𝐶 V
209, 19preqr2 3531 . . . . . . . 8 ({𝐷, B} = {𝐷, 𝐶} → B = 𝐶)
2118, 20syl6bi 152 . . . . . . 7 (A = 𝐷 → ({A, B} = {𝐷, 𝐶} → B = 𝐶))
2221com12 27 . . . . . 6 ({A, B} = {𝐷, 𝐶} → (A = 𝐷B = 𝐶))
2316, 22sylbi 114 . . . . 5 ({A, B} = {𝐶, 𝐷} → (A = 𝐷B = 𝐶))
2423ancld 308 . . . 4 ({A, B} = {𝐶, 𝐷} → (A = 𝐷 → (A = 𝐷 B = 𝐶)))
2514, 24orim12d 699 . . 3 ({A, B} = {𝐶, 𝐷} → ((A = 𝐶 A = 𝐷) → ((A = 𝐶 B = 𝐷) (A = 𝐷 B = 𝐶))))
266, 25mpd 13 . 2 ({A, B} = {𝐶, 𝐷} → ((A = 𝐶 B = 𝐷) (A = 𝐷 B = 𝐶)))
27 preq12 3440 . . 3 ((A = 𝐶 B = 𝐷) → {A, B} = {𝐶, 𝐷})
28 prcom 3437 . . . . 5 {𝐷, B} = {B, 𝐷}
2917, 28syl6eq 2085 . . . 4 (A = 𝐷 → {A, B} = {B, 𝐷})
30 preq1 3438 . . . 4 (B = 𝐶 → {B, 𝐷} = {𝐶, 𝐷})
3129, 30sylan9eq 2089 . . 3 ((A = 𝐷 B = 𝐶) → {A, B} = {𝐶, 𝐷})
3227, 31jaoi 635 . 2 (((A = 𝐶 B = 𝐷) (A = 𝐷 B = 𝐶)) → {A, B} = {𝐶, 𝐷})
3326, 32impbii 117 1 ({A, B} = {𝐶, 𝐷} ↔ ((A = 𝐶 B = 𝐷) (A = 𝐷 B = 𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   wo 628   = wceq 1242   wcel 1390  Vcvv 2551  {cpr 3368
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-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374
This theorem is referenced by:  prel12  3533  opthpr  3534  preq12bg  3535  preqsn  3537  opeqpr  3981  preleq  4233
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