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Theorem heeq2 37092
 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 2610 . 2 𝑅 = 𝑅
2 heeq12 37090 . 2 ((𝑅 = 𝑅𝐴 = 𝐵) → (𝑅 hereditary 𝐴𝑅 hereditary 𝐵))
31, 2mpan 702 1 (𝐴 = 𝐵 → (𝑅 hereditary 𝐴𝑅 hereditary 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   hereditary whe 37086 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-he 37087 This theorem is referenced by: (None)
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