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Mirrors > Home > ILE Home > Th. List > frforeq3 | GIF version |
Description: Equality theorem for the well-founded predicate. (Contributed by Jim Kingdon, 22-Sep-2021.) |
Ref | Expression |
---|---|
frforeq3 | ⊢ (𝑆 = 𝑇 → ( FrFor 𝑅𝐴𝑆 ↔ FrFor 𝑅𝐴𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq2 2221 | . . . . . . 7 ⊢ (𝑆 = 𝑇 → (𝑦 ∈ 𝑆 ↔ 𝑦 ∈ 𝑇)) | |
2 | 1 | imbi2d 229 | . . . . . 6 ⊢ (𝑆 = 𝑇 → ((𝑦𝑅𝑥 → 𝑦 ∈ 𝑆) ↔ (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇))) |
3 | 2 | ralbidv 2457 | . . . . 5 ⊢ (𝑆 = 𝑇 → (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑆) ↔ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇))) |
4 | eleq2 2221 | . . . . 5 ⊢ (𝑆 = 𝑇 → (𝑥 ∈ 𝑆 ↔ 𝑥 ∈ 𝑇)) | |
5 | 3, 4 | imbi12d 233 | . . . 4 ⊢ (𝑆 = 𝑇 → ((∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑆) → 𝑥 ∈ 𝑆) ↔ (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇))) |
6 | 5 | ralbidv 2457 | . . 3 ⊢ (𝑆 = 𝑇 → (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑆) → 𝑥 ∈ 𝑆) ↔ ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇))) |
7 | sseq2 3152 | . . 3 ⊢ (𝑆 = 𝑇 → (𝐴 ⊆ 𝑆 ↔ 𝐴 ⊆ 𝑇)) | |
8 | 6, 7 | imbi12d 233 | . 2 ⊢ (𝑆 = 𝑇 → ((∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑆) → 𝑥 ∈ 𝑆) → 𝐴 ⊆ 𝑆) ↔ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇) → 𝐴 ⊆ 𝑇))) |
9 | df-frfor 4293 | . 2 ⊢ ( FrFor 𝑅𝐴𝑆 ↔ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑆) → 𝑥 ∈ 𝑆) → 𝐴 ⊆ 𝑆)) | |
10 | df-frfor 4293 | . 2 ⊢ ( FrFor 𝑅𝐴𝑇 ↔ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇) → 𝐴 ⊆ 𝑇)) | |
11 | 8, 9, 10 | 3bitr4g 222 | 1 ⊢ (𝑆 = 𝑇 → ( FrFor 𝑅𝐴𝑆 ↔ FrFor 𝑅𝐴𝑇)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1335 ∈ wcel 2128 ∀wral 2435 ⊆ wss 3102 class class class wbr 3967 FrFor wfrfor 4289 |
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-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-11 1486 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-ral 2440 df-in 3108 df-ss 3115 df-frfor 4293 |
This theorem is referenced by: frind 4314 |
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