Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
||
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 2821 | . 2 ⊢ 𝑅 = 𝑅 | |
2 | heeq12 40142 | . 2 ⊢ ((𝑅 = 𝑅 ∧ 𝐴 = 𝐵) → (𝑅 hereditary 𝐴 ↔ 𝑅 hereditary 𝐵)) | |
3 | 1, 2 | mpan 688 | 1 ⊢ (𝐴 = 𝐵 → (𝑅 hereditary 𝐴 ↔ 𝑅 hereditary 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 hereditary whe 40138 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-xp 5561 df-cnv 5563 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-he 40139 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |