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Theorem dfsymrel3 38022
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 38021 . 2 ( SymRel 𝑅 ↔ (𝑅𝑅 ∧ Rel 𝑅))
2 cnvsym 6118 . . 3 (𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥))
32anbi1i 623 . 2 ((𝑅𝑅 ∧ Rel 𝑅) ↔ (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ Rel 𝑅))
41, 3bitri 275 1 ( SymRel 𝑅 ↔ (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ Rel 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wal 1532  wss 3947   class class class wbr 5148  ccnv 5677  Rel wrel 5683   SymRel wsymrel 37660
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-xp 5684  df-rel 5685  df-cnv 5686  df-dm 5688  df-rn 5689  df-res 5690  df-symrel 38016
This theorem is referenced by:  refsymrel3  38040
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