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Theorem elpr2OLD 4148
 Description: Obsolete proof of elpr2 4147 as of 23-Jul-2021. (Contributed by NM, 14-Oct-2005.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
elpr2.1 𝐵 ∈ V
elpr2.2 𝐶 ∈ V
Assertion
Ref Expression
elpr2OLD (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶))

Proof of Theorem elpr2OLD
StepHypRef Expression
1 elpri 4145 . 2 (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵𝐴 = 𝐶))
2 elpr2.1 . . . . . 6 𝐵 ∈ V
3 eleq1 2676 . . . . . 6 (𝐴 = 𝐵 → (𝐴 ∈ V ↔ 𝐵 ∈ V))
42, 3mpbiri 247 . . . . 5 (𝐴 = 𝐵𝐴 ∈ V)
5 elpr2.2 . . . . . 6 𝐶 ∈ V
6 eleq1 2676 . . . . . 6 (𝐴 = 𝐶 → (𝐴 ∈ V ↔ 𝐶 ∈ V))
75, 6mpbiri 247 . . . . 5 (𝐴 = 𝐶𝐴 ∈ V)
84, 7jaoi 393 . . . 4 ((𝐴 = 𝐵𝐴 = 𝐶) → 𝐴 ∈ V)
9 elprg 4144 . . . 4 (𝐴 ∈ V → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
108, 9syl 17 . . 3 ((𝐴 = 𝐵𝐴 = 𝐶) → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
1110ibir 256 . 2 ((𝐴 = 𝐵𝐴 = 𝐶) → 𝐴 ∈ {𝐵, 𝐶})
121, 11impbii 198 1 (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∨ wo 382   = wceq 1475   ∈ wcel 1977  Vcvv 3173  {cpr 4127 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-un 3545  df-sn 4126  df-pr 4128 This theorem is referenced by: (None)
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