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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 7867 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 3779 | . 2 ⊢ (𝑧 = ∅ → ([𝑧 / 𝑥]𝜑 ↔ [∅ / 𝑥]𝜑)) | |
2 | sbequ 2079 | . 2 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
3 | dfsbcq2 3779 | . 2 ⊢ (𝑧 = suc 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [suc 𝑦 / 𝑥]𝜑)) | |
4 | sbequ12r 2240 | . 2 ⊢ (𝑧 = 𝑥 → ([𝑧 / 𝑥]𝜑 ↔ 𝜑)) | |
5 | findes.1 | . 2 ⊢ [∅ / 𝑥]𝜑 | |
6 | nfv 1910 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ ω | |
7 | nfs1v 2146 | . . . . 5 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 | |
8 | nfsbc1v 3796 | . . . . 5 ⊢ Ⅎ𝑥[suc 𝑦 / 𝑥]𝜑 | |
9 | 7, 8 | nfim 1892 | . . . 4 ⊢ Ⅎ𝑥([𝑦 / 𝑥]𝜑 → [suc 𝑦 / 𝑥]𝜑) |
10 | 6, 9 | nfim 1892 | . . 3 ⊢ Ⅎ𝑥(𝑦 ∈ ω → ([𝑦 / 𝑥]𝜑 → [suc 𝑦 / 𝑥]𝜑)) |
11 | eleq1w 2812 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ ω ↔ 𝑦 ∈ ω)) | |
12 | sbequ12 2239 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
13 | suceq 6435 | . . . . . 6 ⊢ (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦) | |
14 | 13 | sbceq1d 3781 | . . . . 5 ⊢ (𝑥 = 𝑦 → ([suc 𝑥 / 𝑥]𝜑 ↔ [suc 𝑦 / 𝑥]𝜑)) |
15 | 12, 14 | imbi12d 344 | . . . 4 ⊢ (𝑥 = 𝑦 → ((𝜑 → [suc 𝑥 / 𝑥]𝜑) ↔ ([𝑦 / 𝑥]𝜑 → [suc 𝑦 / 𝑥]𝜑))) |
16 | 11, 15 | imbi12d 344 | . . 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 2229 | . 2 ⊢ (𝑦 ∈ ω → ([𝑦 / 𝑥]𝜑 → [suc 𝑦 / 𝑥]𝜑)) |
19 | 1, 2, 3, 4, 5, 18 | finds 7904 | 1 ⊢ (𝑥 ∈ ω → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 [wsb 2060 ∈ wcel 2099 [wsbc 3776 ∅c0 4323 suc csuc 6371 ωcom 7870 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-tr 5266 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-om 7871 |
This theorem is referenced by: rdgeqoa 36849 |
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