![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > tfis2f | GIF version |
Description: Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.) |
Ref | Expression |
---|---|
tfis2f.1 | ⊢ Ⅎ𝑥𝜓 |
tfis2f.2 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
tfis2f.3 | ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 𝜓 → 𝜑)) |
Ref | Expression |
---|---|
tfis2f | ⊢ (𝑥 ∈ On → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tfis2f.1 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
2 | tfis2f.2 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
3 | 1, 2 | sbie 1716 | . . . 4 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
4 | 3 | ralbii 2377 | . . 3 ⊢ (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 ↔ ∀𝑦 ∈ 𝑥 𝜓) |
5 | tfis2f.3 | . . 3 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 𝜓 → 𝜑)) | |
6 | 4, 5 | syl5bi 150 | . 2 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑)) |
7 | 6 | tfis 4352 | 1 ⊢ (𝑥 ∈ On → 𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 Ⅎwnf 1390 ∈ wcel 1434 [wsb 1687 ∀wral 2353 Oncon0 4146 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-setind 4308 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-nf 1391 df-sb 1688 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ral 2358 df-rex 2359 df-rab 2362 df-v 2612 df-in 2988 df-ss 2995 df-uni 3622 df-tr 3896 df-iord 4149 df-on 4151 |
This theorem is referenced by: tfis2 4354 tfri3 6037 |
Copyright terms: Public domain | W3C validator |