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Mirrors > Home > ILE Home > Th. List > frforeq1 | GIF version |
Description: Equality theorem for the well-founded predicate. (Contributed by Jim Kingdon, 22-Sep-2021.) |
Ref | Expression |
---|---|
frforeq1 | ⊢ (𝑅 = 𝑆 → ( FrFor 𝑅𝐴𝑇 ↔ FrFor 𝑆𝐴𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq 3766 | . . . . . . 7 ⊢ (𝑅 = 𝑆 → (𝑦𝑅𝑥 ↔ 𝑦𝑆𝑥)) | |
2 | 1 | imbi1d 220 | . . . . . 6 ⊢ (𝑅 = 𝑆 → ((𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) ↔ (𝑦𝑆𝑥 → 𝑦 ∈ 𝑇))) |
3 | 2 | ralbidv 2326 | . . . . 5 ⊢ (𝑅 = 𝑆 → (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) ↔ ∀𝑦 ∈ 𝐴 (𝑦𝑆𝑥 → 𝑦 ∈ 𝑇))) |
4 | 3 | imbi1d 220 | . . . 4 ⊢ (𝑅 = 𝑆 → ((∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇) ↔ (∀𝑦 ∈ 𝐴 (𝑦𝑆𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇))) |
5 | 4 | ralbidv 2326 | . . 3 ⊢ (𝑅 = 𝑆 → (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇) ↔ ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑆𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇))) |
6 | 5 | imbi1d 220 | . 2 ⊢ (𝑅 = 𝑆 → ((∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇) → 𝐴 ⊆ 𝑇) ↔ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑆𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇) → 𝐴 ⊆ 𝑇))) |
7 | df-frfor 4068 | . 2 ⊢ ( FrFor 𝑅𝐴𝑇 ↔ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇) → 𝐴 ⊆ 𝑇)) | |
8 | df-frfor 4068 | . 2 ⊢ ( FrFor 𝑆𝐴𝑇 ↔ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑆𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇) → 𝐴 ⊆ 𝑇)) | |
9 | 6, 7, 8 | 3bitr4g 212 | 1 ⊢ (𝑅 = 𝑆 → ( FrFor 𝑅𝐴𝑇 ↔ FrFor 𝑆𝐴𝑇)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 = wceq 1243 ∈ wcel 1393 ∀wral 2306 ⊆ wss 2917 class class class wbr 3764 FrFor wfrfor 4064 |
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-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-4 1400 ax-17 1419 ax-ial 1427 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-cleq 2033 df-clel 2036 df-ral 2311 df-br 3765 df-frfor 4068 |
This theorem is referenced by: freq1 4081 |
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