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Theorem brresOLD 5562
 Description: Old proof of brres 5560. Obsolete as of 18-Feb-2022. (Contributed by NM, 12-Dec-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
opelresOLD.1 𝐵 ∈ V
Assertion
Ref Expression
brresOLD (𝐴(𝐶𝐷)𝐵 ↔ (𝐴𝐶𝐵𝐴𝐷))

Proof of Theorem brresOLD
StepHypRef Expression
1 opelresOLD.1 . . 3 𝐵 ∈ V
21opelres 5559 . 2 (⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴𝐷))
3 df-br 4805 . 2 (𝐴(𝐶𝐷)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷))
4 df-br 4805 . . 3 (𝐴𝐶𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐶)
54anbi1i 733 . 2 ((𝐴𝐶𝐵𝐴𝐷) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴𝐷))
62, 3, 53bitr4i 292 1 (𝐴(𝐶𝐷)𝐵 ↔ (𝐴𝐶𝐵𝐴𝐷))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∧ wa 383   ∈ wcel 2139  Vcvv 3340  ⟨cop 4327   class class class wbr 4804   ↾ cres 5268 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-res 5278 This theorem is referenced by: (None)
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