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Theorem relcnveq4 36598
Description: Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 28-Apr-2019.)
Assertion
Ref Expression
relcnveq4 (Rel 𝑅 → (𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
Distinct variable group:   𝑥,𝑅,𝑦

Proof of Theorem relcnveq4
StepHypRef Expression
1 relcnveq 36596 . 2 (Rel 𝑅 → (𝑅𝑅𝑅 = 𝑅))
2 relcnveq2 36597 . 2 (Rel 𝑅 → (𝑅 = 𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
31, 2bitrd 278 1 (Rel 𝑅 → (𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1538   = wceq 1540  wss 3898   class class class wbr 5092  ccnv 5619  Rel wrel 5625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pr 5372
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-br 5093  df-opab 5155  df-xp 5626  df-rel 5627  df-cnv 5628
This theorem is referenced by:  dfsymrel5  36827
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