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Mirrors > Home > MPE Home > Th. List > Mathboxes > heeq2 | Structured version Visualization version GIF version |
Description: Equality law for relations being herditary over a class. (Contributed by RP, 27-Mar-2020.) |
Ref | Expression |
---|---|
heeq2 | ⊢ (𝐴 = 𝐵 → (𝑅 hereditary 𝐴 ↔ 𝑅 hereditary 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2738 | . 2 ⊢ 𝑅 = 𝑅 | |
2 | heeq12 41365 | . 2 ⊢ ((𝑅 = 𝑅 ∧ 𝐴 = 𝐵) → (𝑅 hereditary 𝐴 ↔ 𝑅 hereditary 𝐵)) | |
3 | 1, 2 | mpan 687 | 1 ⊢ (𝐴 = 𝐵 → (𝑅 hereditary 𝐴 ↔ 𝑅 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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