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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 1764 | . . . 4 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
4 | 3 | ralbii 2439 | . . 3 ⊢ (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 ↔ ∀𝑦 ∈ 𝑥 𝜓) |
5 | tfis2f.3 | . . 3 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 𝜓 → 𝜑)) | |
6 | 4, 5 | syl5bi 151 | . 2 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑)) |
7 | 6 | tfis 4492 | 1 ⊢ (𝑥 ∈ On → 𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 Ⅎwnf 1436 ∈ wcel 1480 [wsb 1735 ∀wral 2414 Oncon0 4280 |
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 2119 ax-setind 4447 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-in 3072 df-ss 3079 df-uni 3732 df-tr 4022 df-iord 4283 df-on 4285 |
This theorem is referenced by: tfis2 4494 tfri3 6257 |
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