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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 1674 | . . . 4 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
4 | 3 | ralbii 2330 | . . 3 ⊢ (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 ↔ ∀𝑦 ∈ 𝑥 𝜓) |
5 | tfis2f.3 | . . 3 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 𝜓 → 𝜑)) | |
6 | 4, 5 | syl5bi 141 | . 2 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑)) |
7 | 6 | tfis 4306 | 1 ⊢ (𝑥 ∈ On → 𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 Ⅎwnf 1349 ∈ wcel 1393 [wsb 1645 ∀wral 2306 Oncon0 4100 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-setind 4262 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-in 2924 df-ss 2931 df-uni 3581 df-tr 3855 df-iord 4103 df-on 4105 |
This theorem is referenced by: tfis2 4308 tfri3 5953 |
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