![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > finds1 | GIF version |
Description: Principle of Finite Induction (inference schema), using implicit substitutions. The first three hypotheses establish the substitutions we need. The last two are the basis and the induction step. Theorem Schema 22 of [Suppes] p. 136. (Contributed by NM, 22-Mar-2006.) |
Ref | Expression |
---|---|
finds1.1 | ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓)) |
finds1.2 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) |
finds1.3 | ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃)) |
finds1.4 | ⊢ 𝜓 |
finds1.5 | ⊢ (𝑦 ∈ ω → (𝜒 → 𝜃)) |
Ref | Expression |
---|---|
finds1 | ⊢ (𝑥 ∈ ω → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2113 | . 2 ⊢ ∅ = ∅ | |
2 | finds1.1 | . . 3 ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓)) | |
3 | finds1.2 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) | |
4 | finds1.3 | . . 3 ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃)) | |
5 | finds1.4 | . . . 4 ⊢ 𝜓 | |
6 | 5 | a1i 9 | . . 3 ⊢ (∅ = ∅ → 𝜓) |
7 | finds1.5 | . . . 4 ⊢ (𝑦 ∈ ω → (𝜒 → 𝜃)) | |
8 | 7 | a1d 22 | . . 3 ⊢ (𝑦 ∈ ω → (∅ = ∅ → (𝜒 → 𝜃))) |
9 | 2, 3, 4, 6, 8 | finds2 4473 | . 2 ⊢ (𝑥 ∈ ω → (∅ = ∅ → 𝜑)) |
10 | 1, 9 | mpi 15 | 1 ⊢ (𝑥 ∈ ω → 𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1312 ∈ wcel 1461 ∅c0 3327 suc csuc 4245 ωcom 4462 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 ax-nul 4012 ax-pow 4056 ax-pr 4089 ax-un 4313 ax-iinf 4460 |
This theorem depends on definitions: df-bi 116 df-3an 945 df-tru 1315 df-nf 1418 df-sb 1717 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ral 2393 df-rex 2394 df-v 2657 df-dif 3037 df-un 3039 df-in 3041 df-ss 3048 df-nul 3328 df-pw 3476 df-sn 3497 df-pr 3498 df-uni 3701 df-int 3736 df-suc 4251 df-iom 4463 |
This theorem is referenced by: findcard 6733 findcard2 6734 findcard2s 6735 |
Copyright terms: Public domain | W3C validator |