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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 7062 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 3438 | . 2 ⊢ (𝑧 = ∅ → ([𝑧 / 𝑥]𝜑 ↔ [∅ / 𝑥]𝜑)) | |
2 | sbequ 2376 | . 2 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
3 | dfsbcq2 3438 | . 2 ⊢ (𝑧 = suc 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [suc 𝑦 / 𝑥]𝜑)) | |
4 | sbequ12r 2112 | . 2 ⊢ (𝑧 = 𝑥 → ([𝑧 / 𝑥]𝜑 ↔ 𝜑)) | |
5 | findes.1 | . 2 ⊢ [∅ / 𝑥]𝜑 | |
6 | nfv 1843 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ ω | |
7 | nfs1v 2437 | . . . . 5 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 | |
8 | nfsbc1v 3455 | . . . . 5 ⊢ Ⅎ𝑥[suc 𝑦 / 𝑥]𝜑 | |
9 | 7, 8 | nfim 1825 | . . . 4 ⊢ Ⅎ𝑥([𝑦 / 𝑥]𝜑 → [suc 𝑦 / 𝑥]𝜑) |
10 | 6, 9 | nfim 1825 | . . 3 ⊢ Ⅎ𝑥(𝑦 ∈ ω → ([𝑦 / 𝑥]𝜑 → [suc 𝑦 / 𝑥]𝜑)) |
11 | eleq1 2689 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ ω ↔ 𝑦 ∈ ω)) | |
12 | sbequ12 2111 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
13 | suceq 5790 | . . . . . 6 ⊢ (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦) | |
14 | 13 | sbceq1d 3440 | . . . . 5 ⊢ (𝑥 = 𝑦 → ([suc 𝑥 / 𝑥]𝜑 ↔ [suc 𝑦 / 𝑥]𝜑)) |
15 | 12, 14 | imbi12d 334 | . . . 4 ⊢ (𝑥 = 𝑦 → ((𝜑 → [suc 𝑥 / 𝑥]𝜑) ↔ ([𝑦 / 𝑥]𝜑 → [suc 𝑦 / 𝑥]𝜑))) |
16 | 11, 15 | imbi12d 334 | . . 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 2262 | . 2 ⊢ (𝑦 ∈ ω → ([𝑦 / 𝑥]𝜑 → [suc 𝑦 / 𝑥]𝜑)) |
19 | 1, 2, 3, 4, 5, 18 | finds 7092 | 1 ⊢ (𝑥 ∈ ω → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 [wsb 1880 ∈ wcel 1990 [wsbc 3435 ∅c0 3915 suc csuc 5725 ωcom 7065 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-tr 4753 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-om 7066 |
This theorem is referenced by: rdgeqoa 33218 |
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