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

Theorem cnvsymOLDOLD 6113
Description: Obsolete proof of cnvsym 6111 as of 23-Dec-2024. (Contributed by NM, 28-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
cnvsymOLDOLD (𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥))
Distinct variable group:   𝑥,𝑦,𝑅

Proof of Theorem cnvsymOLDOLD
StepHypRef Expression
1 alcom 2157 . 2 (∀𝑦𝑥(⟨𝑦, 𝑥⟩ ∈ 𝑅 → ⟨𝑦, 𝑥⟩ ∈ 𝑅) ↔ ∀𝑥𝑦(⟨𝑦, 𝑥⟩ ∈ 𝑅 → ⟨𝑦, 𝑥⟩ ∈ 𝑅))
2 relcnv 6101 . . 3 Rel 𝑅
3 ssrel 5781 . . 3 (Rel 𝑅 → (𝑅𝑅 ↔ ∀𝑦𝑥(⟨𝑦, 𝑥⟩ ∈ 𝑅 → ⟨𝑦, 𝑥⟩ ∈ 𝑅)))
42, 3ax-mp 5 . 2 (𝑅𝑅 ↔ ∀𝑦𝑥(⟨𝑦, 𝑥⟩ ∈ 𝑅 → ⟨𝑦, 𝑥⟩ ∈ 𝑅))
5 vex 3479 . . . . . 6 𝑦 ∈ V
6 vex 3479 . . . . . 6 𝑥 ∈ V
75, 6brcnv 5881 . . . . 5 (𝑦𝑅𝑥𝑥𝑅𝑦)
8 df-br 5149 . . . . 5 (𝑦𝑅𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝑅)
97, 8bitr3i 277 . . . 4 (𝑥𝑅𝑦 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝑅)
10 df-br 5149 . . . 4 (𝑦𝑅𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝑅)
119, 10imbi12i 351 . . 3 ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝑅 → ⟨𝑦, 𝑥⟩ ∈ 𝑅))
12112albii 1823 . 2 (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ↔ ∀𝑥𝑦(⟨𝑦, 𝑥⟩ ∈ 𝑅 → ⟨𝑦, 𝑥⟩ ∈ 𝑅))
131, 4, 123bitr4i 303 1 (𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1540  wcel 2107  wss 3948  cop 4634   class class class wbr 5148  ccnv 5675  Rel wrel 5681
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-11 2155  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pr 5427
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 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-cnv 5684
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator