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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 2234 | . . . . . . 7 ⊢ (𝑆 = 𝑇 → (𝑦 ∈ 𝑆 ↔ 𝑦 ∈ 𝑇)) | |
2 | 1 | imbi2d 229 | . . . . . 6 ⊢ (𝑆 = 𝑇 → ((𝑦𝑅𝑥 → 𝑦 ∈ 𝑆) ↔ (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇))) |
3 | 2 | ralbidv 2470 | . . . . 5 ⊢ (𝑆 = 𝑇 → (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑆) ↔ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇))) |
4 | eleq2 2234 | . . . . 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 2470 | . . 3 ⊢ (𝑆 = 𝑇 → (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑆) → 𝑥 ∈ 𝑆) ↔ ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇))) |
7 | sseq2 3171 | . . 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 4316 | . 2 ⊢ ( FrFor 𝑅𝐴𝑆 ↔ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑆) → 𝑥 ∈ 𝑆) → 𝐴 ⊆ 𝑆)) | |
10 | df-frfor 4316 | . 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 1348 ∈ wcel 2141 ∀wral 2448 ⊆ wss 3121 class class class wbr 3989 FrFor wfrfor 4312 |
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 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-11 1499 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-ral 2453 df-in 3127 df-ss 3134 df-frfor 4316 |
This theorem is referenced by: frind 4337 |
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