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Theorem brcnvin 36832
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 5157 . 2 (𝐴(𝑅𝑆)𝐵 ↔ (𝐴𝑅𝐵𝐴𝑆𝐵))
2 brcnvg 5835 . . 3 ((𝐴𝑉𝐵𝑊) → (𝐴𝑆𝐵𝐵𝑆𝐴))
32anbi2d 629 . 2 ((𝐴𝑉𝐵𝑊) → ((𝐴𝑅𝐵𝐴𝑆𝐵) ↔ (𝐴𝑅𝐵𝐵𝑆𝐴)))
41, 3bitrid 282 1 ((𝐴𝑉𝐵𝑊) → (𝐴(𝑅𝑆)𝐵 ↔ (𝐴𝑅𝐵𝐵𝑆𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wcel 2106  cin 3909   class class class wbr 5105  ccnv 5632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-br 5106  df-opab 5168  df-cnv 5641
This theorem is referenced by:  dfantisymrel5  37224
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