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Theorem symreleq 35325
 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 5630 . . . 4 (𝑅 = 𝑆𝑅 = 𝑆)
2 id 22 . . . 4 (𝑅 = 𝑆𝑅 = 𝑆)
31, 2sseq12d 3921 . . 3 (𝑅 = 𝑆 → (𝑅𝑅𝑆𝑆))
4 releq 5537 . . 3 (𝑅 = 𝑆 → (Rel 𝑅 ↔ Rel 𝑆))
53, 4anbi12d 630 . 2 (𝑅 = 𝑆 → ((𝑅𝑅 ∧ Rel 𝑅) ↔ (𝑆𝑆 ∧ Rel 𝑆)))
6 dfsymrel2 35316 . 2 ( SymRel 𝑅 ↔ (𝑅𝑅 ∧ Rel 𝑅))
7 dfsymrel2 35316 . 2 ( SymRel 𝑆 ↔ (𝑆𝑆 ∧ Rel 𝑆))
85, 6, 73bitr4g 315 1 (𝑅 = 𝑆 → ( SymRel 𝑅 ↔ SymRel 𝑆))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1522   ⊆ wss 3859  ◡ccnv 5442  Rel wrel 5448   SymRel wsymrel 34997 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pr 5221 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-br 4963  df-opab 5025  df-xp 5449  df-rel 5450  df-cnv 5451  df-dm 5453  df-rn 5454  df-res 5455  df-symrel 35311 This theorem is referenced by:  eqvreleq  35368
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