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Theorem cnvsym 6114
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 2155. (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 6104 . . 3 Rel 𝑅
2 ssrel3 5787 . . 3 (Rel 𝑅 → (𝑅𝑅 ↔ ∀𝑦𝑥(𝑦𝑅𝑥𝑦𝑅𝑥)))
31, 2ax-mp 5 . 2 (𝑅𝑅 ↔ ∀𝑦𝑥(𝑦𝑅𝑥𝑦𝑅𝑥))
4 breq1 5152 . . . 4 (𝑦 = 𝑧 → (𝑦𝑅𝑥𝑧𝑅𝑥))
5 breq1 5152 . . . 4 (𝑦 = 𝑧 → (𝑦𝑅𝑥𝑧𝑅𝑥))
64, 5imbi12d 345 . . 3 (𝑦 = 𝑧 → ((𝑦𝑅𝑥𝑦𝑅𝑥) ↔ (𝑧𝑅𝑥𝑧𝑅𝑥)))
7 breq2 5153 . . . 4 (𝑥 = 𝑧 → (𝑦𝑅𝑥𝑦𝑅𝑧))
8 breq2 5153 . . . 4 (𝑥 = 𝑧 → (𝑦𝑅𝑥𝑦𝑅𝑧))
97, 8imbi12d 345 . . 3 (𝑥 = 𝑧 → ((𝑦𝑅𝑥𝑦𝑅𝑥) ↔ (𝑦𝑅𝑧𝑦𝑅𝑧)))
106, 9alcomw 2048 . 2 (∀𝑦𝑥(𝑦𝑅𝑥𝑦𝑅𝑥) ↔ ∀𝑥𝑦(𝑦𝑅𝑥𝑦𝑅𝑥))
11 vex 3479 . . . . 5 𝑦 ∈ V
12 vex 3479 . . . . 5 𝑥 ∈ V
1311, 12brcnv 5883 . . . 4 (𝑦𝑅𝑥𝑥𝑅𝑦)
1413imbi1i 350 . . 3 ((𝑦𝑅𝑥𝑦𝑅𝑥) ↔ (𝑥𝑅𝑦𝑦𝑅𝑥))
15142albii 1823 . 2 (∀𝑥𝑦(𝑦𝑅𝑥𝑦𝑅𝑥) ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥))
163, 10, 153bitri 297 1 (𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1540  wss 3949   class class class wbr 5149  ccnv 5676  Rel wrel 5682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-cnv 5685
This theorem is referenced by:  dfer2  8704  relcnveq3  37190  relcnveq  37191  relcnveq2  37192  cnvcosseq  37307  symrelcoss2  37336  elrelscnveq3  37361  elrelscnveq  37362  elrelscnveq2  37363  dfsymrels3  37416  dfsymrel3  37420  symrefref3  37434  refsymrels3  37436  elrefsymrels3  37440  dfeqvrels3  37459
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