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Theorem symreleq 36672
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 5782 . . . 4 (𝑅 = 𝑆𝑅 = 𝑆)
2 id 22 . . . 4 (𝑅 = 𝑆𝑅 = 𝑆)
31, 2sseq12d 3954 . . 3 (𝑅 = 𝑆 → (𝑅𝑅𝑆𝑆))
4 releq 5687 . . 3 (𝑅 = 𝑆 → (Rel 𝑅 ↔ Rel 𝑆))
53, 4anbi12d 631 . 2 (𝑅 = 𝑆 → ((𝑅𝑅 ∧ Rel 𝑅) ↔ (𝑆𝑆 ∧ Rel 𝑆)))
6 dfsymrel2 36663 . 2 ( SymRel 𝑅 ↔ (𝑅𝑅 ∧ Rel 𝑅))
7 dfsymrel2 36663 . 2 ( SymRel 𝑆 ↔ (𝑆𝑆 ∧ Rel 𝑆))
85, 6, 73bitr4g 314 1 (𝑅 = 𝑆 → ( SymRel 𝑅 ↔ SymRel 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1539  wss 3887  ccnv 5588  Rel wrel 5594   SymRel wsymrel 36345
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-dm 5599  df-rn 5600  df-res 5601  df-symrel 36658
This theorem is referenced by:  eqvreleq  36715
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