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Theorem cnvsymOLDOLD 6134
Description: Obsolete version of cnvsym 6132 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 2159 . 2 (∀𝑦𝑥(⟨𝑦, 𝑥⟩ ∈ 𝑅 → ⟨𝑦, 𝑥⟩ ∈ 𝑅) ↔ ∀𝑥𝑦(⟨𝑦, 𝑥⟩ ∈ 𝑅 → ⟨𝑦, 𝑥⟩ ∈ 𝑅))
2 relcnv 6122 . . 3 Rel 𝑅
3 ssrel 5792 . . 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 3484 . . . . . 6 𝑦 ∈ V
6 vex 3484 . . . . . 6 𝑥 ∈ V
75, 6brcnv 5893 . . . . 5 (𝑦𝑅𝑥𝑥𝑅𝑦)
8 df-br 5144 . . . . 5 (𝑦𝑅𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝑅)
97, 8bitr3i 277 . . . 4 (𝑥𝑅𝑦 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝑅)
10 df-br 5144 . . . 4 (𝑦𝑅𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝑅)
119, 10imbi12i 350 . . 3 ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝑅 → ⟨𝑦, 𝑥⟩ ∈ 𝑅))
12112albii 1820 . 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 206  wal 1538  wcel 2108  wss 3951  cop 4632   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-11 2157  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: (None)
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