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Theorem relcnveq2 37851
Description: Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 28-Apr-2019.)
Assertion
Ref Expression
relcnveq2 (Rel 𝑅 → (𝑅 = 𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
Distinct variable group:   𝑥,𝑅,𝑦

Proof of Theorem relcnveq2
StepHypRef Expression
1 cnvsym 6113 . . . 4 (𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥))
21a1i 11 . . 3 (Rel 𝑅 → (𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
3 dfrel2 6188 . . . . . . 7 (Rel 𝑅𝑅 = 𝑅)
43biimpi 215 . . . . . 6 (Rel 𝑅𝑅 = 𝑅)
54sseq1d 4004 . . . . 5 (Rel 𝑅 → (𝑅𝑅𝑅𝑅))
6 cnvsym 6113 . . . . 5 (𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥))
75, 6bitr3di 285 . . . 4 (Rel 𝑅 → (𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
8 relbrcnvg 6104 . . . . . 6 (Rel 𝑅 → (𝑥𝑅𝑦𝑦𝑅𝑥))
9 relbrcnvg 6104 . . . . . 6 (Rel 𝑅 → (𝑦𝑅𝑥𝑥𝑅𝑦))
108, 9imbi12d 343 . . . . 5 (Rel 𝑅 → ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ (𝑦𝑅𝑥𝑥𝑅𝑦)))
11102albidv 1918 . . . 4 (Rel 𝑅 → (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ↔ ∀𝑥𝑦(𝑦𝑅𝑥𝑥𝑅𝑦)))
127, 11bitrd 278 . . 3 (Rel 𝑅 → (𝑅𝑅 ↔ ∀𝑥𝑦(𝑦𝑅𝑥𝑥𝑅𝑦)))
132, 12anbi12d 630 . 2 (Rel 𝑅 → ((𝑅𝑅𝑅𝑅) ↔ (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ ∀𝑥𝑦(𝑦𝑅𝑥𝑥𝑅𝑦))))
14 eqss 3988 . 2 (𝑅 = 𝑅 ↔ (𝑅𝑅𝑅𝑅))
15 2albiim 1885 . 2 (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ↔ (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ ∀𝑥𝑦(𝑦𝑅𝑥𝑥𝑅𝑦)))
1613, 14, 153bitr4g 313 1 (Rel 𝑅 → (𝑅 = 𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wal 1531   = wceq 1533  wss 3939   class class class wbr 5143  ccnv 5671  Rel wrel 5677
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 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
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 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5144  df-opab 5206  df-xp 5678  df-rel 5679  df-cnv 5680
This theorem is referenced by:  relcnveq4  37852
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