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Theorem eqvreleq 37093
Description: Equality theorem for equivalence relation. (Contributed by Peter Mazsa, 19-Apr-2020.) (Revised by Peter Mazsa, 23-Sep-2021.)
Assertion
Ref Expression
eqvreleq (𝑅 = 𝑆 → ( EqvRel 𝑅 ↔ EqvRel 𝑆))

Proof of Theorem eqvreleq
StepHypRef Expression
1 refreleq 37012 . . 3 (𝑅 = 𝑆 → ( RefRel 𝑅 ↔ RefRel 𝑆))
2 symreleq 37049 . . 3 (𝑅 = 𝑆 → ( SymRel 𝑅 ↔ SymRel 𝑆))
3 trreleq 37073 . . 3 (𝑅 = 𝑆 → ( TrRel 𝑅 ↔ TrRel 𝑆))
41, 2, 33anbi123d 1437 . 2 (𝑅 = 𝑆 → (( RefRel 𝑅 ∧ SymRel 𝑅 ∧ TrRel 𝑅) ↔ ( RefRel 𝑆 ∧ SymRel 𝑆 ∧ TrRel 𝑆)))
5 df-eqvrel 37076 . 2 ( EqvRel 𝑅 ↔ ( RefRel 𝑅 ∧ SymRel 𝑅 ∧ TrRel 𝑅))
6 df-eqvrel 37076 . 2 ( EqvRel 𝑆 ↔ ( RefRel 𝑆 ∧ SymRel 𝑆 ∧ TrRel 𝑆))
74, 5, 63bitr4g 314 1 (𝑅 = 𝑆 → ( EqvRel 𝑅 ↔ EqvRel 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  w3a 1088   = wceq 1542   RefRel wrefrel 36669   SymRel wsymrel 36675   TrRel wtrrel 36678   EqvRel weqvrel 36680
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-refrel 37003  df-symrel 37035  df-trrel 37065  df-eqvrel 37076
This theorem is referenced by:  eqvreleqi  37094  eqvreleqd  37095  erALTVeq1  37160
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