Theorem releq 2477

Description: Equality theorem for relation predicate.

Assertion

Ref	Expression
releq	⊢ (A = B → (Rel A ↔ Rel B))

Proof of Theorem releq

Step	Hyp	Ref	Expression
1		sseq1 1521	. 2 ⊢ (A = B → (A ⊆ (V × V) ↔ B ⊆ (V × V)))
2		df-rel 2425	. 2 ⊢ (Rel A ↔ A ⊆ (V × V))
3		df-rel 2425	. 2 ⊢ (Rel B ↔ B ⊆ (V × V))
4	1, 2, 3	3bitr4g 428	1 ⊢ (A = B → (Rel A ↔ Rel B))

Syntax hints:   → wi 2   ↔ wb 127   = wceq 1091  Vcvv 1348   ⊆ wss 1487   × cxp 2408  Rel wrel 2415

This theorem is referenced by:  reli 2500  rele 2501  relcnv 2624  relco 2658  dfrel2 2660  tfrlem6 2954  reloprab 3022  reldmoprab 3034  ndmoprcl 3058  relen 3277  reldom 3278  relsdom 3279  mapdom2lem 3388  aceq3lem 3555

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-in 1491  df-ss 1492  df-rel 2425

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