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Mirrors > Home > MPE Home > Th. List > nnindd | Structured version Visualization version GIF version |
Description: Principle of Mathematical Induction (inference schema) on integers, a deduction version. (Contributed by Thierry Arnoux, 19-Jul-2020.) |
Ref | Expression |
---|---|
nnindd.1 | ⊢ (𝑥 = 1 → (𝜓 ↔ 𝜒)) |
nnindd.2 | ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜃)) |
nnindd.3 | ⊢ (𝑥 = (𝑦 + 1) → (𝜓 ↔ 𝜏)) |
nnindd.4 | ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜂)) |
nnindd.5 | ⊢ (𝜑 → 𝜒) |
nnindd.6 | ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ 𝜃) → 𝜏) |
Ref | Expression |
---|---|
nnindd | ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → 𝜂) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnindd.1 | . . . 4 ⊢ (𝑥 = 1 → (𝜓 ↔ 𝜒)) | |
2 | 1 | imbi2d 345 | . . 3 ⊢ (𝑥 = 1 → ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒))) |
3 | nnindd.2 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜃)) | |
4 | 3 | imbi2d 345 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝜑 → 𝜓) ↔ (𝜑 → 𝜃))) |
5 | nnindd.3 | . . . 4 ⊢ (𝑥 = (𝑦 + 1) → (𝜓 ↔ 𝜏)) | |
6 | 5 | imbi2d 345 | . . 3 ⊢ (𝑥 = (𝑦 + 1) → ((𝜑 → 𝜓) ↔ (𝜑 → 𝜏))) |
7 | nnindd.4 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜂)) | |
8 | 7 | imbi2d 345 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝜑 → 𝜓) ↔ (𝜑 → 𝜂))) |
9 | nnindd.5 | . . 3 ⊢ (𝜑 → 𝜒) | |
10 | nnindd.6 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ 𝜃) → 𝜏) | |
11 | 10 | ex 417 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ) → (𝜃 → 𝜏)) |
12 | 11 | expcom 418 | . . . 4 ⊢ (𝑦 ∈ ℕ → (𝜑 → (𝜃 → 𝜏))) |
13 | 12 | a2d 29 | . . 3 ⊢ (𝑦 ∈ ℕ → ((𝜑 → 𝜃) → (𝜑 → 𝜏))) |
14 | 2, 4, 6, 8, 9, 13 | nnind 11677 | . 2 ⊢ (𝐴 ∈ ℕ → (𝜑 → 𝜂)) |
15 | 14 | impcom 412 | 1 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → 𝜂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1539 ∈ wcel 2112 (class class class)co 7143 1c1 10561 + caddc 10563 ℕcn 11659 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-sep 5162 ax-nul 5169 ax-pr 5291 ax-un 7452 ax-1cn 10618 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-ral 3073 df-rex 3074 df-reu 3075 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-pss 3873 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-tp 4520 df-op 4522 df-uni 4792 df-iun 4878 df-br 5026 df-opab 5088 df-mpt 5106 df-tr 5132 df-id 5423 df-eprel 5428 df-po 5436 df-so 5437 df-fr 5476 df-we 5478 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-pred 6119 df-ord 6165 df-on 6166 df-lim 6167 df-suc 6168 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-ov 7146 df-om 7573 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-nn 11660 |
This theorem is referenced by: fzto1st 30881 psgnfzto1st 30883 fiunelros 31646 fsuppind 39769 |
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