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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 35920 | . . 3 ⊢ (𝑅 = 𝑆 → ( RefRel 𝑅 ↔ RefRel 𝑆)) | |
2 | symreleq 35954 | . . 3 ⊢ (𝑅 = 𝑆 → ( SymRel 𝑅 ↔ SymRel 𝑆)) | |
3 | trreleq 35978 | . . 3 ⊢ (𝑅 = 𝑆 → ( TrRel 𝑅 ↔ TrRel 𝑆)) | |
4 | 1, 2, 3 | 3anbi123d 1433 | . 2 ⊢ (𝑅 = 𝑆 → (( RefRel 𝑅 ∧ SymRel 𝑅 ∧ TrRel 𝑅) ↔ ( RefRel 𝑆 ∧ SymRel 𝑆 ∧ TrRel 𝑆))) |
5 | df-eqvrel 35980 | . 2 ⊢ ( EqvRel 𝑅 ↔ ( RefRel 𝑅 ∧ SymRel 𝑅 ∧ TrRel 𝑅)) | |
6 | df-eqvrel 35980 | . 2 ⊢ ( EqvRel 𝑆 ↔ ( RefRel 𝑆 ∧ SymRel 𝑆 ∧ TrRel 𝑆)) | |
7 | 4, 5, 6 | 3bitr4g 317 | 1 ⊢ (𝑅 = 𝑆 → ( EqvRel 𝑅 ↔ EqvRel 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ w3a 1084 = wceq 1538 RefRel wrefrel 35619 SymRel wsymrel 35625 TrRel wtrrel 35628 EqvRel weqvrel 35630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-refrel 35912 df-symrel 35940 df-trrel 35970 df-eqvrel 35980 |
This theorem is referenced by: eqvreleqi 35998 eqvreleqd 35999 erALTVeq1 36063 |
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