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Theorem symreleq 37049
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 5834 . . . 4 (𝑅 = 𝑆𝑅 = 𝑆)
2 id 22 . . . 4 (𝑅 = 𝑆𝑅 = 𝑆)
31, 2sseq12d 3982 . . 3 (𝑅 = 𝑆 → (𝑅𝑅𝑆𝑆))
4 releq 5737 . . 3 (𝑅 = 𝑆 → (Rel 𝑅 ↔ Rel 𝑆))
53, 4anbi12d 632 . 2 (𝑅 = 𝑆 → ((𝑅𝑅 ∧ Rel 𝑅) ↔ (𝑆𝑆 ∧ Rel 𝑆)))
6 dfsymrel2 37040 . 2 ( SymRel 𝑅 ↔ (𝑅𝑅 ∧ Rel 𝑅))
7 dfsymrel2 37040 . 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 397   = wceq 1542  wss 3915  ccnv 5637  Rel wrel 5643   SymRel wsymrel 36675
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-xp 5644  df-rel 5645  df-cnv 5646  df-dm 5648  df-rn 5649  df-res 5650  df-symrel 37035
This theorem is referenced by:  eqvreleq  37093
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