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Mirrors > Home > MPE Home > Th. List > Mathboxes > eqvreleq | Structured version Visualization version GIF version |
Description: Equality theorem for equivalence relation. (Contributed by Peter Mazsa, 19-Apr-2020.) (Revised by Peter Mazsa, 23-Sep-2021.) |
Ref | Expression |
---|---|
eqvreleq | ⊢ (𝑅 = 𝑆 → ( EqvRel 𝑅 ↔ EqvRel 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | refreleq 34764 | . . 3 ⊢ (𝑅 = 𝑆 → ( RefRel 𝑅 ↔ RefRel 𝑆)) | |
2 | symreleq 34798 | . . 3 ⊢ (𝑅 = 𝑆 → ( SymRel 𝑅 ↔ SymRel 𝑆)) | |
3 | trreleq 34822 | . . 3 ⊢ (𝑅 = 𝑆 → ( TrRel 𝑅 ↔ TrRel 𝑆)) | |
4 | 1, 2, 3 | 3anbi123d 1561 | . 2 ⊢ (𝑅 = 𝑆 → (( RefRel 𝑅 ∧ SymRel 𝑅 ∧ TrRel 𝑅) ↔ ( RefRel 𝑆 ∧ SymRel 𝑆 ∧ TrRel 𝑆))) |
5 | df-eqvrel 34824 | . 2 ⊢ ( EqvRel 𝑅 ↔ ( RefRel 𝑅 ∧ SymRel 𝑅 ∧ TrRel 𝑅)) | |
6 | df-eqvrel 34824 | . 2 ⊢ ( EqvRel 𝑆 ↔ ( RefRel 𝑆 ∧ SymRel 𝑆 ∧ TrRel 𝑆)) | |
7 | 4, 5, 6 | 3bitr4g 306 | 1 ⊢ (𝑅 = 𝑆 → ( EqvRel 𝑅 ↔ EqvRel 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ w3a 1108 = wceq 1653 RefRel wrefrel 34475 SymRel wsymrel 34481 TrRel wtrrel 34484 EqvRel weqvrel 34486 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pr 5097 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-op 4375 df-br 4844 df-opab 4906 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-refrel 34756 df-symrel 34784 df-trrel 34814 df-eqvrel 34824 |
This theorem is referenced by: eqvreleqi 34839 eqvreleqd 34840 |
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