Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  eqvreleq Structured version   Visualization version   GIF version

Theorem eqvreleq 36715
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 36638 . . 3 (𝑅 = 𝑆 → ( RefRel 𝑅 ↔ RefRel 𝑆))
2 symreleq 36672 . . 3 (𝑅 = 𝑆 → ( SymRel 𝑅 ↔ SymRel 𝑆))
3 trreleq 36696 . . 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 36698 . 2 ( EqvRel 𝑅 ↔ ( RefRel 𝑅 ∧ SymRel 𝑅 ∧ TrRel 𝑅))
6 df-eqvrel 36698 . 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 1086   = wceq 1539   RefRel wrefrel 36339   SymRel wsymrel 36345   TrRel wtrrel 36348   EqvRel weqvrel 36350
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-refrel 36630  df-symrel 36658  df-trrel 36688  df-eqvrel 36698
This theorem is referenced by:  eqvreleqi  36716  eqvreleqd  36717  erALTVeq1  36781
  Copyright terms: Public domain W3C validator