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Theorem prel12OLD 4515
Description: Obsolete as of 9-Sep-2021. Use prel12g 4531 instead. (Contributed by NM, 17-Oct-1996.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
preqr1.a 𝐴 ∈ V
preqr1.b 𝐵 ∈ V
preq12b.c 𝐶 ∈ V
preq12b.d 𝐷 ∈ V
Assertion
Ref Expression
prel12OLD 𝐴 = 𝐵 → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 ∈ {𝐶, 𝐷} ∧ 𝐵 ∈ {𝐶, 𝐷})))

Proof of Theorem prel12OLD
StepHypRef Expression
1 preqr1.a . . . . 5 𝐴 ∈ V
21prid1 4434 . . . 4 𝐴 ∈ {𝐴, 𝐵}
3 eleq2 2839 . . . 4 ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 ∈ {𝐴, 𝐵} ↔ 𝐴 ∈ {𝐶, 𝐷}))
42, 3mpbii 223 . . 3 ({𝐴, 𝐵} = {𝐶, 𝐷} → 𝐴 ∈ {𝐶, 𝐷})
5 preqr1.b . . . . 5 𝐵 ∈ V
65prid2 4435 . . . 4 𝐵 ∈ {𝐴, 𝐵}
7 eleq2 2839 . . . 4 ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐵 ∈ {𝐴, 𝐵} ↔ 𝐵 ∈ {𝐶, 𝐷}))
86, 7mpbii 223 . . 3 ({𝐴, 𝐵} = {𝐶, 𝐷} → 𝐵 ∈ {𝐶, 𝐷})
94, 8jca 501 . 2 ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 ∈ {𝐶, 𝐷} ∧ 𝐵 ∈ {𝐶, 𝐷}))
101elpr 4339 . . . 4 (𝐴 ∈ {𝐶, 𝐷} ↔ (𝐴 = 𝐶𝐴 = 𝐷))
11 eqeq2 2782 . . . . . . . . . . . 12 (𝐵 = 𝐷 → (𝐴 = 𝐵𝐴 = 𝐷))
1211notbid 307 . . . . . . . . . . 11 (𝐵 = 𝐷 → (¬ 𝐴 = 𝐵 ↔ ¬ 𝐴 = 𝐷))
13 orel2 877 . . . . . . . . . . 11 𝐴 = 𝐷 → ((𝐴 = 𝐶𝐴 = 𝐷) → 𝐴 = 𝐶))
1412, 13syl6bi 243 . . . . . . . . . 10 (𝐵 = 𝐷 → (¬ 𝐴 = 𝐵 → ((𝐴 = 𝐶𝐴 = 𝐷) → 𝐴 = 𝐶)))
1514impd 396 . . . . . . . . 9 (𝐵 = 𝐷 → ((¬ 𝐴 = 𝐵 ∧ (𝐴 = 𝐶𝐴 = 𝐷)) → 𝐴 = 𝐶))
1615com12 32 . . . . . . . 8 ((¬ 𝐴 = 𝐵 ∧ (𝐴 = 𝐶𝐴 = 𝐷)) → (𝐵 = 𝐷𝐴 = 𝐶))
1716ancrd 541 . . . . . . 7 ((¬ 𝐴 = 𝐵 ∧ (𝐴 = 𝐶𝐴 = 𝐷)) → (𝐵 = 𝐷 → (𝐴 = 𝐶𝐵 = 𝐷)))
18 eqeq2 2782 . . . . . . . . . . . 12 (𝐵 = 𝐶 → (𝐴 = 𝐵𝐴 = 𝐶))
1918notbid 307 . . . . . . . . . . 11 (𝐵 = 𝐶 → (¬ 𝐴 = 𝐵 ↔ ¬ 𝐴 = 𝐶))
20 orel1 875 . . . . . . . . . . 11 𝐴 = 𝐶 → ((𝐴 = 𝐶𝐴 = 𝐷) → 𝐴 = 𝐷))
2119, 20syl6bi 243 . . . . . . . . . 10 (𝐵 = 𝐶 → (¬ 𝐴 = 𝐵 → ((𝐴 = 𝐶𝐴 = 𝐷) → 𝐴 = 𝐷)))
2221impd 396 . . . . . . . . 9 (𝐵 = 𝐶 → ((¬ 𝐴 = 𝐵 ∧ (𝐴 = 𝐶𝐴 = 𝐷)) → 𝐴 = 𝐷))
2322com12 32 . . . . . . . 8 ((¬ 𝐴 = 𝐵 ∧ (𝐴 = 𝐶𝐴 = 𝐷)) → (𝐵 = 𝐶𝐴 = 𝐷))
2423ancrd 541 . . . . . . 7 ((¬ 𝐴 = 𝐵 ∧ (𝐴 = 𝐶𝐴 = 𝐷)) → (𝐵 = 𝐶 → (𝐴 = 𝐷𝐵 = 𝐶)))
2517, 24orim12d 949 . . . . . 6 ((¬ 𝐴 = 𝐵 ∧ (𝐴 = 𝐶𝐴 = 𝐷)) → ((𝐵 = 𝐷𝐵 = 𝐶) → ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
265elpr 4339 . . . . . . 7 (𝐵 ∈ {𝐶, 𝐷} ↔ (𝐵 = 𝐶𝐵 = 𝐷))
27 orcom 859 . . . . . . 7 ((𝐵 = 𝐶𝐵 = 𝐷) ↔ (𝐵 = 𝐷𝐵 = 𝐶))
2826, 27bitri 264 . . . . . 6 (𝐵 ∈ {𝐶, 𝐷} ↔ (𝐵 = 𝐷𝐵 = 𝐶))
29 preq12b.c . . . . . . 7 𝐶 ∈ V
30 preq12b.d . . . . . . 7 𝐷 ∈ V
311, 5, 29, 30preq12b 4514 . . . . . 6 ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)))
3225, 28, 313imtr4g 285 . . . . 5 ((¬ 𝐴 = 𝐵 ∧ (𝐴 = 𝐶𝐴 = 𝐷)) → (𝐵 ∈ {𝐶, 𝐷} → {𝐴, 𝐵} = {𝐶, 𝐷}))
3332ex 397 . . . 4 𝐴 = 𝐵 → ((𝐴 = 𝐶𝐴 = 𝐷) → (𝐵 ∈ {𝐶, 𝐷} → {𝐴, 𝐵} = {𝐶, 𝐷})))
3410, 33syl5bi 232 . . 3 𝐴 = 𝐵 → (𝐴 ∈ {𝐶, 𝐷} → (𝐵 ∈ {𝐶, 𝐷} → {𝐴, 𝐵} = {𝐶, 𝐷})))
3534impd 396 . 2 𝐴 = 𝐵 → ((𝐴 ∈ {𝐶, 𝐷} ∧ 𝐵 ∈ {𝐶, 𝐷}) → {𝐴, 𝐵} = {𝐶, 𝐷}))
369, 35impbid2 216 1 𝐴 = 𝐵 → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 ∈ {𝐶, 𝐷} ∧ 𝐵 ∈ {𝐶, 𝐷})))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  wo 836   = wceq 1631  wcel 2145  Vcvv 3351  {cpr 4319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-un 3728  df-sn 4318  df-pr 4320
This theorem is referenced by:  prel12gOLD  4519  dfac2OLD  9159
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