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Theorem brcnvin 37240
Description: Intersection with a converse, binary relation. (Contributed by Peter Mazsa, 24-Mar-2024.)
Assertion
Ref Expression
brcnvin ((𝐴𝑉𝐵𝑊) → (𝐴(𝑅𝑆)𝐵 ↔ (𝐴𝑅𝐵𝐵𝑆𝐴)))

Proof of Theorem brcnvin
StepHypRef Expression
1 brin 5201 . 2 (𝐴(𝑅𝑆)𝐵 ↔ (𝐴𝑅𝐵𝐴𝑆𝐵))
2 brcnvg 5880 . . 3 ((𝐴𝑉𝐵𝑊) → (𝐴𝑆𝐵𝐵𝑆𝐴))
32anbi2d 630 . 2 ((𝐴𝑉𝐵𝑊) → ((𝐴𝑅𝐵𝐴𝑆𝐵) ↔ (𝐴𝑅𝐵𝐵𝑆𝐴)))
41, 3bitrid 283 1 ((𝐴𝑉𝐵𝑊) → (𝐴(𝑅𝑆)𝐵 ↔ (𝐴𝑅𝐵𝐵𝑆𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wcel 2107  cin 3948   class class class wbr 5149  ccnv 5676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-cnv 5685
This theorem is referenced by:  dfantisymrel5  37632
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