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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 341 | . . 3 ⊢ (𝑥 = 1 → ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒))) |
3 | nnindd.2 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜃)) | |
4 | 3 | imbi2d 341 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝜑 → 𝜓) ↔ (𝜑 → 𝜃))) |
5 | nnindd.3 | . . . 4 ⊢ (𝑥 = (𝑦 + 1) → (𝜓 ↔ 𝜏)) | |
6 | 5 | imbi2d 341 | . . 3 ⊢ (𝑥 = (𝑦 + 1) → ((𝜑 → 𝜓) ↔ (𝜑 → 𝜏))) |
7 | nnindd.4 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜂)) | |
8 | 7 | imbi2d 341 | . . 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 414 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ) → (𝜃 → 𝜏)) |
12 | 11 | expcom 415 | . . . 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 12179 | . 2 ⊢ (𝐴 ∈ ℕ → (𝜑 → 𝜂)) |
15 | 14 | impcom 409 | 1 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → 𝜂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 (class class class)co 7361 1c1 11060 + caddc 11062 ℕcn 12161 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 ax-un 7676 ax-1cn 11117 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7364 df-om 7807 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-nn 12162 |
This theorem is referenced by: fzto1st 32008 psgnfzto1st 32010 fiunelros 32837 fsuppind 40812 sn-nnne0 40964 renegmulnnass 40969 |
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