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Theorem cnvsym 6132
Description: Two ways of saying a relation is symmetric. Similar to definition of symmetry in [Schechter] p. 51. (Contributed by NM, 28-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Proof shortened by SN, 23-Dec-2024.) Avoid ax-11 2157. (Revised by BTernaryTau, 29-Dec-2024.)
Assertion
Ref Expression
cnvsym (𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥))
Distinct variable group:   𝑥,𝑦,𝑅

Proof of Theorem cnvsym
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 relcnv 6122 . . 3 Rel 𝑅
2 ssrel3 5796 . . 3 (Rel 𝑅 → (𝑅𝑅 ↔ ∀𝑦𝑥(𝑦𝑅𝑥𝑦𝑅𝑥)))
31, 2ax-mp 5 . 2 (𝑅𝑅 ↔ ∀𝑦𝑥(𝑦𝑅𝑥𝑦𝑅𝑥))
4 breq1 5146 . . . 4 (𝑦 = 𝑧 → (𝑦𝑅𝑥𝑧𝑅𝑥))
5 breq1 5146 . . . 4 (𝑦 = 𝑧 → (𝑦𝑅𝑥𝑧𝑅𝑥))
64, 5imbi12d 344 . . 3 (𝑦 = 𝑧 → ((𝑦𝑅𝑥𝑦𝑅𝑥) ↔ (𝑧𝑅𝑥𝑧𝑅𝑥)))
7 breq2 5147 . . . 4 (𝑥 = 𝑧 → (𝑦𝑅𝑥𝑦𝑅𝑧))
8 breq2 5147 . . . 4 (𝑥 = 𝑧 → (𝑦𝑅𝑥𝑦𝑅𝑧))
97, 8imbi12d 344 . . 3 (𝑥 = 𝑧 → ((𝑦𝑅𝑥𝑦𝑅𝑥) ↔ (𝑦𝑅𝑧𝑦𝑅𝑧)))
106, 9alcomw 2044 . 2 (∀𝑦𝑥(𝑦𝑅𝑥𝑦𝑅𝑥) ↔ ∀𝑥𝑦(𝑦𝑅𝑥𝑦𝑅𝑥))
11 vex 3484 . . . . 5 𝑦 ∈ V
12 vex 3484 . . . . 5 𝑥 ∈ V
1311, 12brcnv 5893 . . . 4 (𝑦𝑅𝑥𝑥𝑅𝑦)
1413imbi1i 349 . . 3 ((𝑦𝑅𝑥𝑦𝑅𝑥) ↔ (𝑥𝑅𝑦𝑦𝑅𝑥))
15142albii 1820 . 2 (∀𝑥𝑦(𝑦𝑅𝑥𝑦𝑅𝑥) ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥))
163, 10, 153bitri 297 1 (𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1538  wss 3951   class class class wbr 5143  ccnv 5684  Rel wrel 5690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693
This theorem is referenced by:  dfer2  8746  relcnveq3  38322  relcnveq  38323  relcnveq2  38324  cnvcosseq  38438  symrelcoss2  38467  elrelscnveq3  38492  elrelscnveq  38493  elrelscnveq2  38494  dfsymrels3  38547  dfsymrel3  38551  symrefref3  38565  refsymrels3  38567  elrefsymrels3  38571  dfeqvrels3  38590
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