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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 329 | . . 3 ⊢ (𝑥 = 1 → ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒))) |
3 | nnindd.2 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜃)) | |
4 | 3 | imbi2d 329 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝜑 → 𝜓) ↔ (𝜑 → 𝜃))) |
5 | nnindd.3 | . . . 4 ⊢ (𝑥 = (𝑦 + 1) → (𝜓 ↔ 𝜏)) | |
6 | 5 | imbi2d 329 | . . 3 ⊢ (𝑥 = (𝑦 + 1) → ((𝜑 → 𝜓) ↔ (𝜑 → 𝜏))) |
7 | nnindd.4 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜂)) | |
8 | 7 | imbi2d 329 | . . 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 449 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ) → (𝜃 → 𝜏)) |
12 | 11 | expcom 450 | . . . 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 10915 | . 2 ⊢ (𝐴 ∈ ℕ → (𝜑 → 𝜂)) |
15 | 14 | impcom 445 | 1 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → 𝜂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 (class class class)co 6549 1c1 9816 + caddc 9818 ℕcn 10897 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-1cn 9873 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-nn 10898 |
This theorem is referenced by: fzto1st 29184 psgnfzto1st 29186 fiunelros 29564 |
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