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Theorem releq 4422
 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 2966 . 2 (𝐴 = 𝐵 → (𝐴 ⊆ (V × V) ↔ 𝐵 ⊆ (V × V)))
2 df-rel 4352 . 2 (Rel 𝐴𝐴 ⊆ (V × V))
3 df-rel 4352 . 2 (Rel 𝐵𝐵 ⊆ (V × V))
41, 2, 33bitr4g 212 1 (𝐴 = 𝐵 → (Rel 𝐴 ↔ Rel 𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1243  Vcvv 2557   ⊆ wss 2917   × cxp 4343  Rel wrel 4350 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-in 2924  df-ss 2931  df-rel 4352 This theorem is referenced by:  releqi  4423  releqd  4424  dfrel2  4771  tposfn2  5881  ereq1  6113
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