MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnvsymOLD Structured version   Visualization version   GIF version

Theorem cnvsymOLD 6114
Description: Obsolete proof of cnvsym 6113 as of 29-Dec-2024. (Contributed by NM, 28-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Proof shortened by SN, 23-Dec-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
cnvsymOLD (𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥))
Distinct variable group:   𝑥,𝑦,𝑅

Proof of Theorem cnvsymOLD
StepHypRef Expression
1 relcnv 6103 . . 3 Rel 𝑅
2 ssrel3 5782 . . 3 (Rel 𝑅 → (𝑅𝑅 ↔ ∀𝑦𝑥(𝑦𝑅𝑥𝑦𝑅𝑥)))
31, 2ax-mp 5 . 2 (𝑅𝑅 ↔ ∀𝑦𝑥(𝑦𝑅𝑥𝑦𝑅𝑥))
4 alcom 2148 . 2 (∀𝑦𝑥(𝑦𝑅𝑥𝑦𝑅𝑥) ↔ ∀𝑥𝑦(𝑦𝑅𝑥𝑦𝑅𝑥))
5 vex 3467 . . . . 5 𝑦 ∈ V
6 vex 3467 . . . . 5 𝑥 ∈ V
75, 6brcnv 5879 . . . 4 (𝑦𝑅𝑥𝑥𝑅𝑦)
87imbi1i 348 . . 3 ((𝑦𝑅𝑥𝑦𝑅𝑥) ↔ (𝑥𝑅𝑦𝑦𝑅𝑥))
982albii 1814 . 2 (∀𝑥𝑦(𝑦𝑅𝑥𝑦𝑅𝑥) ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥))
103, 4, 93bitri 296 1 (𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1531  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-11 2146  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-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: (None)
  Copyright terms: Public domain W3C validator