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Theorem symreleq 34665
Description: Equality theorem for symmetric relation. (Contributed by Peter Mazsa, 15-Apr-2019.) (Revised by Peter Mazsa, 23-Sep-2021.)
Assertion
Ref Expression
symreleq (𝑅 = 𝑆 → ( SymRel 𝑅 ↔ SymRel 𝑆))

Proof of Theorem symreleq
StepHypRef Expression
1 cnveq 5463 . . . 4 (𝑅 = 𝑆𝑅 = 𝑆)
2 id 22 . . . 4 (𝑅 = 𝑆𝑅 = 𝑆)
31, 2sseq12d 3793 . . 3 (𝑅 = 𝑆 → (𝑅𝑅𝑆𝑆))
4 releq 5370 . . 3 (𝑅 = 𝑆 → (Rel 𝑅 ↔ Rel 𝑆))
53, 4anbi12d 624 . 2 (𝑅 = 𝑆 → ((𝑅𝑅 ∧ Rel 𝑅) ↔ (𝑆𝑆 ∧ Rel 𝑆)))
6 dfsymrel2 34656 . 2 ( SymRel 𝑅 ↔ (𝑅𝑅 ∧ Rel 𝑅))
7 dfsymrel2 34656 . 2 ( SymRel 𝑆 ↔ (𝑆𝑆 ∧ Rel 𝑆))
85, 6, 73bitr4g 305 1 (𝑅 = 𝑆 → ( SymRel 𝑅 ↔ SymRel 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wss 3731  ccnv 5275  Rel wrel 5281   SymRel wsymrel 34348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-sep 4940  ax-nul 4948  ax-pr 5061
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2564  df-eu 2581  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3351  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-nul 4079  df-if 4243  df-sn 4334  df-pr 4336  df-op 4340  df-br 4809  df-opab 4871  df-xp 5282  df-rel 5283  df-cnv 5284  df-dm 5286  df-rn 5287  df-res 5288  df-symrel 34651
This theorem is referenced by:  eqvreleq  34705
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