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Theorem dfsymrel3 38062
Description: Alternate definition of the symmetric relation predicate. (Contributed by Peter Mazsa, 21-Apr-2019.) (Revised by Peter Mazsa, 17-Aug-2021.)
Assertion
Ref Expression
dfsymrel3 ( SymRel 𝑅 ↔ (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ Rel 𝑅))
Distinct variable group:   𝑥,𝑅,𝑦

Proof of Theorem dfsymrel3
StepHypRef Expression
1 dfsymrel2 38061 . 2 ( SymRel 𝑅 ↔ (𝑅𝑅 ∧ Rel 𝑅))
2 cnvsym 6123 . . 3 (𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥))
32anbi1i 622 . 2 ((𝑅𝑅 ∧ Rel 𝑅) ↔ (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ Rel 𝑅))
41, 3bitri 274 1 ( SymRel 𝑅 ↔ (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ Rel 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wal 1531  wss 3949   class class class wbr 5152  ccnv 5681  Rel wrel 5687   SymRel wsymrel 37701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-xp 5688  df-rel 5689  df-cnv 5690  df-dm 5692  df-rn 5693  df-res 5694  df-symrel 38056
This theorem is referenced by:  refsymrel3  38080
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