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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 4251 | . 2 ⊢ (x ∈ On → φ) |
5 | 1, 4 | vtoclga 2613 | 1 ⊢ (A ∈ On → χ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 = wceq 1242 ∈ wcel 1390 ∀wral 2300 Oncon0 4066 |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-setind 4220 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-rab 2309 df-v 2553 df-in 2918 df-ss 2925 df-uni 3572 df-tr 3846 df-iord 4069 df-on 4071 |
This theorem is referenced by: tfisi 4253 tfrlemi1 5887 rdgon 5913 |
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