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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 35640 | . . 3 ⊢ (𝑅 = 𝑆 → ( RefRel 𝑅 ↔ RefRel 𝑆)) | |
2 | symreleq 35674 | . . 3 ⊢ (𝑅 = 𝑆 → ( SymRel 𝑅 ↔ SymRel 𝑆)) | |
3 | trreleq 35698 | . . 3 ⊢ (𝑅 = 𝑆 → ( TrRel 𝑅 ↔ TrRel 𝑆)) | |
4 | 1, 2, 3 | 3anbi123d 1427 | . 2 ⊢ (𝑅 = 𝑆 → (( RefRel 𝑅 ∧ SymRel 𝑅 ∧ TrRel 𝑅) ↔ ( RefRel 𝑆 ∧ SymRel 𝑆 ∧ TrRel 𝑆))) |
5 | df-eqvrel 35700 | . 2 ⊢ ( EqvRel 𝑅 ↔ ( RefRel 𝑅 ∧ SymRel 𝑅 ∧ TrRel 𝑅)) | |
6 | df-eqvrel 35700 | . 2 ⊢ ( EqvRel 𝑆 ↔ ( RefRel 𝑆 ∧ SymRel 𝑆 ∧ TrRel 𝑆)) | |
7 | 4, 5, 6 | 3bitr4g 315 | 1 ⊢ (𝑅 = 𝑆 → ( EqvRel 𝑅 ↔ EqvRel 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ w3a 1079 = wceq 1528 RefRel wrefrel 35340 SymRel wsymrel 35346 TrRel wtrrel 35349 EqvRel weqvrel 35351 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-refrel 35632 df-symrel 35660 df-trrel 35690 df-eqvrel 35700 |
This theorem is referenced by: eqvreleqi 35718 eqvreleqd 35719 erALTVeq1 35783 |
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