![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 7342 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 3655 | . 2 ⊢ (𝑧 = ∅ → ([𝑧 / 𝑥]𝜑 ↔ [∅ / 𝑥]𝜑)) | |
2 | sbequ 2452 | . 2 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
3 | dfsbcq2 3655 | . 2 ⊢ (𝑧 = suc 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [suc 𝑦 / 𝑥]𝜑)) | |
4 | sbequ12r 2230 | . 2 ⊢ (𝑧 = 𝑥 → ([𝑧 / 𝑥]𝜑 ↔ 𝜑)) | |
5 | findes.1 | . 2 ⊢ [∅ / 𝑥]𝜑 | |
6 | nfv 1957 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ ω | |
7 | nfs1v 2254 | . . . . 5 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 | |
8 | nfsbc1v 3672 | . . . . 5 ⊢ Ⅎ𝑥[suc 𝑦 / 𝑥]𝜑 | |
9 | 7, 8 | nfim 1943 | . . . 4 ⊢ Ⅎ𝑥([𝑦 / 𝑥]𝜑 → [suc 𝑦 / 𝑥]𝜑) |
10 | 6, 9 | nfim 1943 | . . 3 ⊢ Ⅎ𝑥(𝑦 ∈ ω → ([𝑦 / 𝑥]𝜑 → [suc 𝑦 / 𝑥]𝜑)) |
11 | eleq1w 2842 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ ω ↔ 𝑦 ∈ ω)) | |
12 | sbequ12 2229 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
13 | suceq 6043 | . . . . . 6 ⊢ (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦) | |
14 | 13 | sbceq1d 3657 | . . . . 5 ⊢ (𝑥 = 𝑦 → ([suc 𝑥 / 𝑥]𝜑 ↔ [suc 𝑦 / 𝑥]𝜑)) |
15 | 12, 14 | imbi12d 336 | . . . 4 ⊢ (𝑥 = 𝑦 → ((𝜑 → [suc 𝑥 / 𝑥]𝜑) ↔ ([𝑦 / 𝑥]𝜑 → [suc 𝑦 / 𝑥]𝜑))) |
16 | 11, 15 | imbi12d 336 | . . 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 | chvar 2360 | . 2 ⊢ (𝑦 ∈ ω → ([𝑦 / 𝑥]𝜑 → [suc 𝑦 / 𝑥]𝜑)) |
19 | 1, 2, 3, 4, 5, 18 | finds 7372 | 1 ⊢ (𝑥 ∈ ω → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 [wsb 2011 ∈ wcel 2107 [wsbc 3652 ∅c0 4141 suc csuc 5980 ωcom 7345 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pr 5140 ax-un 7228 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-br 4889 df-opab 4951 df-tr 4990 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-om 7346 |
This theorem is referenced by: rdgeqoa 33816 cnfinltrel 33839 |
Copyright terms: Public domain | W3C validator |