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Mirrors > Home > MPE Home > Th. List > findes | Structured version Visualization version GIF version |
Description: Finite induction with explicit substitution. The first hypothesis is the basis and the second is the induction step. Theorem Schema 22 of [Suppes] p. 136. See tfindes 7571 for the transfinite version. This is an alternative for Metamath 100 proof #74. (Contributed by Raph Levien, 9-Jul-2003.) |
Ref | Expression |
---|---|
findes.1 | ⊢ [∅ / 𝑥]𝜑 |
findes.2 | ⊢ (𝑥 ∈ ω → (𝜑 → [suc 𝑥 / 𝑥]𝜑)) |
Ref | Expression |
---|---|
findes | ⊢ (𝑥 ∈ ω → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsbcq2 3775 | . 2 ⊢ (𝑧 = ∅ → ([𝑧 / 𝑥]𝜑 ↔ [∅ / 𝑥]𝜑)) | |
2 | sbequ 2086 | . 2 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
3 | dfsbcq2 3775 | . 2 ⊢ (𝑧 = suc 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [suc 𝑦 / 𝑥]𝜑)) | |
4 | sbequ12r 2249 | . 2 ⊢ (𝑧 = 𝑥 → ([𝑧 / 𝑥]𝜑 ↔ 𝜑)) | |
5 | findes.1 | . 2 ⊢ [∅ / 𝑥]𝜑 | |
6 | nfv 1911 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ ω | |
7 | nfs1v 2269 | . . . . 5 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 | |
8 | nfsbc1v 3792 | . . . . 5 ⊢ Ⅎ𝑥[suc 𝑦 / 𝑥]𝜑 | |
9 | 7, 8 | nfim 1893 | . . . 4 ⊢ Ⅎ𝑥([𝑦 / 𝑥]𝜑 → [suc 𝑦 / 𝑥]𝜑) |
10 | 6, 9 | nfim 1893 | . . 3 ⊢ Ⅎ𝑥(𝑦 ∈ ω → ([𝑦 / 𝑥]𝜑 → [suc 𝑦 / 𝑥]𝜑)) |
11 | eleq1w 2895 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ ω ↔ 𝑦 ∈ ω)) | |
12 | sbequ12 2248 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
13 | suceq 6251 | . . . . . 6 ⊢ (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦) | |
14 | 13 | sbceq1d 3777 | . . . . 5 ⊢ (𝑥 = 𝑦 → ([suc 𝑥 / 𝑥]𝜑 ↔ [suc 𝑦 / 𝑥]𝜑)) |
15 | 12, 14 | imbi12d 347 | . . . 4 ⊢ (𝑥 = 𝑦 → ((𝜑 → [suc 𝑥 / 𝑥]𝜑) ↔ ([𝑦 / 𝑥]𝜑 → [suc 𝑦 / 𝑥]𝜑))) |
16 | 11, 15 | imbi12d 347 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ ω → (𝜑 → [suc 𝑥 / 𝑥]𝜑)) ↔ (𝑦 ∈ ω → ([𝑦 / 𝑥]𝜑 → [suc 𝑦 / 𝑥]𝜑)))) |
17 | findes.2 | . . 3 ⊢ (𝑥 ∈ ω → (𝜑 → [suc 𝑥 / 𝑥]𝜑)) | |
18 | 10, 16, 17 | chvarfv 2237 | . 2 ⊢ (𝑦 ∈ ω → ([𝑦 / 𝑥]𝜑 → [suc 𝑦 / 𝑥]𝜑)) |
19 | 1, 2, 3, 4, 5, 18 | finds 7602 | 1 ⊢ (𝑥 ∈ ω → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 [wsb 2065 ∈ wcel 2110 [wsbc 3772 ∅c0 4291 suc csuc 6188 ωcom 7574 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-tr 5166 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-om 7575 |
This theorem is referenced by: rdgeqoa 34645 |
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