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 38313
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 38232 . . 3 (𝑅 = 𝑆 → ( RefRel 𝑅 ↔ RefRel 𝑆))
2 symreleq 38269 . . 3 (𝑅 = 𝑆 → ( SymRel 𝑅 ↔ SymRel 𝑆))
3 trreleq 38293 . . 3 (𝑅 = 𝑆 → ( TrRel 𝑅 ↔ TrRel 𝑆))
41, 2, 33anbi123d 1433 . 2 (𝑅 = 𝑆 → (( RefRel 𝑅 ∧ SymRel 𝑅 ∧ TrRel 𝑅) ↔ ( RefRel 𝑆 ∧ SymRel 𝑆 ∧ TrRel 𝑆)))
5 df-eqvrel 38296 . 2 ( EqvRel 𝑅 ↔ ( RefRel 𝑅 ∧ SymRel 𝑅 ∧ TrRel 𝑅))
6 df-eqvrel 38296 . 2 ( EqvRel 𝑆 ↔ ( RefRel 𝑆 ∧ SymRel 𝑆 ∧ TrRel 𝑆))
74, 5, 63bitr4g 313 1 (𝑅 = 𝑆 → ( EqvRel 𝑅 ↔ EqvRel 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  w3a 1084   = wceq 1534   RefRel wrefrel 37895   SymRel wsymrel 37901   TrRel wtrrel 37904   EqvRel weqvrel 37906
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pr 5425
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5146  df-opab 5208  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-refrel 38223  df-symrel 38255  df-trrel 38285  df-eqvrel 38296
This theorem is referenced by:  eqvreleqi  38314  eqvreleqd  38315  erALTVeq1  38380
  Copyright terms: Public domain W3C validator