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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 1801 | . . . 4 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
4 | 3 | ralbii 2493 | . . 3 ⊢ (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 ↔ ∀𝑦 ∈ 𝑥 𝜓) |
5 | tfis2f.3 | . . 3 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 𝜓 → 𝜑)) | |
6 | 4, 5 | biimtrid 152 | . 2 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑)) |
7 | 6 | tfis 4594 | 1 ⊢ (𝑥 ∈ On → 𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 Ⅎwnf 1470 [wsb 1772 ∈ wcel 2158 ∀wral 2465 Oncon0 4375 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 ax-setind 4548 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-rab 2474 df-v 2751 df-in 3147 df-ss 3154 df-uni 3822 df-tr 4114 df-iord 4378 df-on 4380 |
This theorem is referenced by: tfis2 4596 tfri3 6381 |
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