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Theorem releq 5645
 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 3991 . 2 (𝐴 = 𝐵 → (𝐴 ⊆ (V × V) ↔ 𝐵 ⊆ (V × V)))
2 df-rel 5556 . 2 (Rel 𝐴𝐴 ⊆ (V × V))
3 df-rel 5556 . 2 (Rel 𝐵𝐵 ⊆ (V × V))
41, 2, 33bitr4g 316 1 (𝐴 = 𝐵 → (Rel 𝐴 ↔ Rel 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   = wceq 1533  Vcvv 3494   ⊆ wss 3935   × cxp 5547  Rel wrel 5554 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-in 3942  df-ss 3951  df-rel 5556 This theorem is referenced by:  releqi  5646  releqd  5647  relsnb  5669  dfrel2  6040  tposfn2  7908  ereq1  8290  isps  17806  isdir  17836  fpwrelmapffslem  30462  bnj1321  32294  refreleq  35754  symreleq  35788  trreleq  35812  prtlem12  35997  relintabex  39934  clrellem  39975  clcnvlem  39976
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