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Theorem heeq2 41367
Description: Equality law for relations being herditary over a class. (Contributed by RP, 27-Mar-2020.)
Assertion
Ref Expression
heeq2 (𝐴 = 𝐵 → (𝑅 hereditary 𝐴𝑅 hereditary 𝐵))

Proof of Theorem heeq2
StepHypRef Expression
1 eqid 2738 . 2 𝑅 = 𝑅
2 heeq12 41365 . 2 ((𝑅 = 𝑅𝐴 = 𝐵) → (𝑅 hereditary 𝐴𝑅 hereditary 𝐵))
31, 2mpan 687 1 (𝐴 = 𝐵 → (𝑅 hereditary 𝐴𝑅 hereditary 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1539   hereditary whe 41361
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3431  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5074  df-opab 5136  df-xp 5590  df-cnv 5592  df-dm 5594  df-rn 5595  df-res 5596  df-ima 5597  df-he 41362
This theorem is referenced by: (None)
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