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Mirrors > Home > ILE Home > Th. List > frforeq2 | GIF version |
Description: Equality theorem for the well-founded predicate. (Contributed by Jim Kingdon, 22-Sep-2021.) |
Ref | Expression |
---|---|
frforeq2 | ⊢ (𝐴 = 𝐵 → ( FrFor 𝑅𝐴𝑇 ↔ FrFor 𝑅𝐵𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | raleq 2626 | . . . . 5 ⊢ (𝐴 = 𝐵 → (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) ↔ ∀𝑦 ∈ 𝐵 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇))) | |
2 | 1 | imbi1d 230 | . . . 4 ⊢ (𝐴 = 𝐵 → ((∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇) ↔ (∀𝑦 ∈ 𝐵 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇))) |
3 | 2 | raleqbi1dv 2634 | . . 3 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇) ↔ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇))) |
4 | sseq1 3120 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ⊆ 𝑇 ↔ 𝐵 ⊆ 𝑇)) | |
5 | 3, 4 | imbi12d 233 | . 2 ⊢ (𝐴 = 𝐵 → ((∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇) → 𝐴 ⊆ 𝑇) ↔ (∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇) → 𝐵 ⊆ 𝑇))) |
6 | df-frfor 4253 | . 2 ⊢ ( FrFor 𝑅𝐴𝑇 ↔ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇) → 𝐴 ⊆ 𝑇)) | |
7 | df-frfor 4253 | . 2 ⊢ ( FrFor 𝑅𝐵𝑇 ↔ (∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇) → 𝐵 ⊆ 𝑇)) | |
8 | 5, 6, 7 | 3bitr4g 222 | 1 ⊢ (𝐴 = 𝐵 → ( FrFor 𝑅𝐴𝑇 ↔ FrFor 𝑅𝐵𝑇)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1331 ∈ wcel 1480 ∀wral 2416 ⊆ wss 3071 class class class wbr 3929 FrFor wfrfor 4249 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-in 3077 df-ss 3084 df-frfor 4253 |
This theorem is referenced by: freq2 4268 |
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