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Theorem releq 5114
Description: Equality theorem for the relation predicate. (Contributed by NM, 1-Aug-1994.)
Assertion
Ref Expression
releq (𝐴 = 𝐵 → (Rel 𝐴 ↔ Rel 𝐵))

Proof of Theorem releq
StepHypRef Expression
1 sseq1 3589 . 2 (𝐴 = 𝐵 → (𝐴 ⊆ (V × V) ↔ 𝐵 ⊆ (V × V)))
2 df-rel 5035 . 2 (Rel 𝐴𝐴 ⊆ (V × V))
3 df-rel 5035 . 2 (Rel 𝐵𝐵 ⊆ (V × V))
41, 2, 33bitr4g 302 1 (𝐴 = 𝐵 → (Rel 𝐴 ↔ Rel 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195   = wceq 1475  Vcvv 3173  wss 3540   × cxp 5026  Rel wrel 5033
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-in 3547  df-ss 3554  df-rel 5035
This theorem is referenced by:  releqi  5115  releqd  5116  dfrel2  5488  tposfn2  7239  ereq1  7614  isps  16974  isdir  17004  fpwrelmapffslem  28689  bnj1321  30143  frrlem6  30827  prtlem12  32964  relintabex  36700  clrellem  36742  clcnvlem  36743
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