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Theorem eqvreleq 38584
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 38503 . . 3 (𝑅 = 𝑆 → ( RefRel 𝑅 ↔ RefRel 𝑆))
2 symreleq 38540 . . 3 (𝑅 = 𝑆 → ( SymRel 𝑅 ↔ SymRel 𝑆))
3 trreleq 38564 . . 3 (𝑅 = 𝑆 → ( TrRel 𝑅 ↔ TrRel 𝑆))
41, 2, 33anbi123d 1435 . 2 (𝑅 = 𝑆 → (( RefRel 𝑅 ∧ SymRel 𝑅 ∧ TrRel 𝑅) ↔ ( RefRel 𝑆 ∧ SymRel 𝑆 ∧ TrRel 𝑆)))
5 df-eqvrel 38567 . 2 ( EqvRel 𝑅 ↔ ( RefRel 𝑅 ∧ SymRel 𝑅 ∧ TrRel 𝑅))
6 df-eqvrel 38567 . 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 206  w3a 1086   = wceq 1537   RefRel wrefrel 38168   SymRel wsymrel 38174   TrRel wtrrel 38177   EqvRel weqvrel 38179
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-refrel 38494  df-symrel 38526  df-trrel 38556  df-eqvrel 38567
This theorem is referenced by:  eqvreleqi  38585  eqvreleqd  38586  erALTVeq1  38651
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