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Theorem eqvreleq 35873
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 35796 . . 3 (𝑅 = 𝑆 → ( RefRel 𝑅 ↔ RefRel 𝑆))
2 symreleq 35830 . . 3 (𝑅 = 𝑆 → ( SymRel 𝑅 ↔ SymRel 𝑆))
3 trreleq 35854 . . 3 (𝑅 = 𝑆 → ( TrRel 𝑅 ↔ TrRel 𝑆))
41, 2, 33anbi123d 1432 . 2 (𝑅 = 𝑆 → (( RefRel 𝑅 ∧ SymRel 𝑅 ∧ TrRel 𝑅) ↔ ( RefRel 𝑆 ∧ SymRel 𝑆 ∧ TrRel 𝑆)))
5 df-eqvrel 35856 . 2 ( EqvRel 𝑅 ↔ ( RefRel 𝑅 ∧ SymRel 𝑅 ∧ TrRel 𝑅))
6 df-eqvrel 35856 . 2 ( EqvRel 𝑆 ↔ ( RefRel 𝑆 ∧ SymRel 𝑆 ∧ TrRel 𝑆))
74, 5, 63bitr4g 316 1 (𝑅 = 𝑆 → ( EqvRel 𝑅 ↔ EqvRel 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  w3a 1083   = wceq 1537   RefRel wrefrel 35495   SymRel wsymrel 35501   TrRel wtrrel 35504   EqvRel weqvrel 35506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pr 5306
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-sn 4544  df-pr 4546  df-op 4550  df-br 5043  df-opab 5105  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-refrel 35788  df-symrel 35816  df-trrel 35846  df-eqvrel 35856
This theorem is referenced by:  eqvreleqi  35874  eqvreleqd  35875  erALTVeq1  35939
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