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Mirrors > Home > ILE Home > Th. List > tfis3 | GIF version |
Description: Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 4-Nov-2003.) |
Ref | Expression |
---|---|
tfis3.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
tfis3.2 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒)) |
tfis3.3 | ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 𝜓 → 𝜑)) |
Ref | Expression |
---|---|
tfis3 | ⊢ (𝐴 ∈ On → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tfis3.2 | . 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒)) | |
2 | tfis3.1 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
3 | tfis3.3 | . . 3 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 𝜓 → 𝜑)) | |
4 | 2, 3 | tfis2 4499 | . 2 ⊢ (𝑥 ∈ On → 𝜑) |
5 | 1, 4 | vtoclga 2752 | 1 ⊢ (𝐴 ∈ On → 𝜒) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1331 ∈ wcel 1480 ∀wral 2416 Oncon0 4285 |
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 ax-setind 4452 |
This theorem depends on definitions: df-bi 116 df-3an 964 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-rex 2422 df-rab 2425 df-v 2688 df-in 3077 df-ss 3084 df-uni 3737 df-tr 4027 df-iord 4288 df-on 4290 |
This theorem is referenced by: tfisi 4501 tfrlemi1 6229 tfr1onlemaccex 6245 tfrcllemaccex 6258 tfrcl 6261 |
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