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Theorem heeq1 37592
Description: Equality law for relations being herditary over a class. (Contributed by RP, 27-Mar-2020.)
Assertion
Ref Expression
heeq1 (𝑅 = 𝑆 → (𝑅 hereditary 𝐴𝑆 hereditary 𝐴))

Proof of Theorem heeq1
StepHypRef Expression
1 eqid 2621 . 2 𝐴 = 𝐴
2 heeq12 37591 . 2 ((𝑅 = 𝑆𝐴 = 𝐴) → (𝑅 hereditary 𝐴𝑆 hereditary 𝐴))
31, 2mpan2 706 1 (𝑅 = 𝑆 → (𝑅 hereditary 𝐴𝑆 hereditary 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1480   hereditary whe 37587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-br 4624  df-opab 4684  df-xp 5090  df-cnv 5092  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-he 37588
This theorem is referenced by:  0heALT  37598
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