ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  frforeq1 GIF version

Theorem frforeq1 4106
Description: Equality theorem for the well-founded predicate. (Contributed by Jim Kingdon, 22-Sep-2021.)
Assertion
Ref Expression
frforeq1 (𝑅 = 𝑆 → ( FrFor 𝑅𝐴𝑇 ↔ FrFor 𝑆𝐴𝑇))

Proof of Theorem frforeq1
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq 3795 . . . . . . 7 (𝑅 = 𝑆 → (𝑦𝑅𝑥𝑦𝑆𝑥))
21imbi1d 229 . . . . . 6 (𝑅 = 𝑆 → ((𝑦𝑅𝑥𝑦𝑇) ↔ (𝑦𝑆𝑥𝑦𝑇)))
32ralbidv 2369 . . . . 5 (𝑅 = 𝑆 → (∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑇) ↔ ∀𝑦𝐴 (𝑦𝑆𝑥𝑦𝑇)))
43imbi1d 229 . . . 4 (𝑅 = 𝑆 → ((∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑇) → 𝑥𝑇) ↔ (∀𝑦𝐴 (𝑦𝑆𝑥𝑦𝑇) → 𝑥𝑇)))
54ralbidv 2369 . . 3 (𝑅 = 𝑆 → (∀𝑥𝐴 (∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑇) → 𝑥𝑇) ↔ ∀𝑥𝐴 (∀𝑦𝐴 (𝑦𝑆𝑥𝑦𝑇) → 𝑥𝑇)))
65imbi1d 229 . 2 (𝑅 = 𝑆 → ((∀𝑥𝐴 (∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑇) → 𝑥𝑇) → 𝐴𝑇) ↔ (∀𝑥𝐴 (∀𝑦𝐴 (𝑦𝑆𝑥𝑦𝑇) → 𝑥𝑇) → 𝐴𝑇)))
7 df-frfor 4094 . 2 ( FrFor 𝑅𝐴𝑇 ↔ (∀𝑥𝐴 (∀𝑦𝐴 (𝑦𝑅𝑥𝑦𝑇) → 𝑥𝑇) → 𝐴𝑇))
8 df-frfor 4094 . 2 ( FrFor 𝑆𝐴𝑇 ↔ (∀𝑥𝐴 (∀𝑦𝐴 (𝑦𝑆𝑥𝑦𝑇) → 𝑥𝑇) → 𝐴𝑇))
96, 7, 83bitr4g 221 1 (𝑅 = 𝑆 → ( FrFor 𝑅𝐴𝑇 ↔ FrFor 𝑆𝐴𝑇))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1285  wcel 1434  wral 2349  wss 2974   class class class wbr 3793   FrFor wfrfor 4090
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-cleq 2075  df-clel 2078  df-ral 2354  df-br 3794  df-frfor 4094
This theorem is referenced by:  freq1  4107
  Copyright terms: Public domain W3C validator