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Theorem cnvssb 39813
Description: Subclass theorem for converse. (Contributed by RP, 22-Oct-2020.)
Assertion
Ref Expression
cnvssb (Rel 𝐴 → (𝐴𝐵𝐴𝐵))

Proof of Theorem cnvssb
StepHypRef Expression
1 cnvss 5741 . 2 (𝐴𝐵𝐴𝐵)
2 cnvss 5741 . . 3 (𝐴𝐵𝐴𝐵)
3 dfrel2 6043 . . . . . . . 8 (Rel 𝐴𝐴 = 𝐴)
43biimpi 217 . . . . . . 7 (Rel 𝐴𝐴 = 𝐴)
54eqcomd 2831 . . . . . 6 (Rel 𝐴𝐴 = 𝐴)
65adantr 481 . . . . 5 ((Rel 𝐴𝐴𝐵) → 𝐴 = 𝐴)
7 id 22 . . . . . . 7 (𝐴𝐵𝐴𝐵)
8 cnvcnvss 6048 . . . . . . 7 𝐵𝐵
97, 8syl6ss 3982 . . . . . 6 (𝐴𝐵𝐴𝐵)
109adantl 482 . . . . 5 ((Rel 𝐴𝐴𝐵) → 𝐴𝐵)
116, 10eqsstrd 4008 . . . 4 ((Rel 𝐴𝐴𝐵) → 𝐴𝐵)
1211ex 413 . . 3 (Rel 𝐴 → (𝐴𝐵𝐴𝐵))
132, 12syl5 34 . 2 (Rel 𝐴 → (𝐴𝐵𝐴𝐵))
141, 13impbid2 227 1 (Rel 𝐴 → (𝐴𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wss 3939  ccnv 5552  Rel wrel 5558
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-rab 3151  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-br 5063  df-opab 5125  df-xp 5559  df-rel 5560  df-cnv 5561
This theorem is referenced by: (None)
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