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Theorem brinxp2OLD 5427
 Description: Obsolete version of brinxp2 5426 as of 18-Sep-2022. (Contributed by NM, 3-Mar-2007.) (Revised by Mario Carneiro, 26-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
brinxp2OLD (𝐴(𝑅 ∩ (𝐶 × 𝐷))𝐵 ↔ (𝐴𝐶𝐵𝐷𝐴𝑅𝐵))

Proof of Theorem brinxp2OLD
StepHypRef Expression
1 brin 4938 . 2 (𝐴(𝑅 ∩ (𝐶 × 𝐷))𝐵 ↔ (𝐴𝑅𝐵𝐴(𝐶 × 𝐷)𝐵))
2 ancom 454 . 2 ((𝐴𝑅𝐵𝐴(𝐶 × 𝐷)𝐵) ↔ (𝐴(𝐶 × 𝐷)𝐵𝐴𝑅𝐵))
3 brxp 5401 . . . 4 (𝐴(𝐶 × 𝐷)𝐵 ↔ (𝐴𝐶𝐵𝐷))
43anbi1i 617 . . 3 ((𝐴(𝐶 × 𝐷)𝐵𝐴𝑅𝐵) ↔ ((𝐴𝐶𝐵𝐷) ∧ 𝐴𝑅𝐵))
5 df-3an 1073 . . 3 ((𝐴𝐶𝐵𝐷𝐴𝑅𝐵) ↔ ((𝐴𝐶𝐵𝐷) ∧ 𝐴𝑅𝐵))
64, 5bitr4i 270 . 2 ((𝐴(𝐶 × 𝐷)𝐵𝐴𝑅𝐵) ↔ (𝐴𝐶𝐵𝐷𝐴𝑅𝐵))
71, 2, 63bitri 289 1 (𝐴(𝑅 ∩ (𝐶 × 𝐷))𝐵 ↔ (𝐴𝐶𝐵𝐷𝐴𝑅𝐵))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198   ∧ wa 386   ∧ w3a 1071   ∈ wcel 2106   ∩ cin 3790   class class class wbr 4886   × cxp 5353 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pr 5138 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-br 4887  df-opab 4949  df-xp 5361 This theorem is referenced by: (None)
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