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Theorem opelresOLD2 5610
 Description: Old proof of opelresi 5606. Obsolete as of 24-Sep-2022. (Contributed by NM, 13-Nov-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
opelresOLD2.1 𝐵 ∈ V
Assertion
Ref Expression
opelresOLD2 (⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴𝐷))

Proof of Theorem opelresOLD2
StepHypRef Expression
1 opelresOLD2.1 . 2 𝐵 ∈ V
2 opelresgOLD2 5608 . 2 (𝐵 ∈ V → (⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴𝐷)))
31, 2ax-mp 5 1 (⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴𝐷))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198   ∧ wa 385   ∈ wcel 2157  Vcvv 3383  ⟨cop 4372   ↾ cres 5312 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2375  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pr 5095 This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-opab 4904  df-xp 5316  df-res 5322 This theorem is referenced by:  brresOLD  5613  opelresgOLD  5614  elresOLD  5644
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