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Mirrors > Home > MPE Home > Th. List > tfindes | Structured version Visualization version GIF version |
Description: Transfinite Induction with explicit substitution. The first hypothesis is the basis, the second is the induction step for successors, and the third is the induction step for limit ordinals. Theorem Schema 4 of [Suppes] p. 197. (Contributed by NM, 5-Mar-2004.) |
Ref | Expression |
---|---|
tfindes.1 | ⊢ [∅ / 𝑥]𝜑 |
tfindes.2 | ⊢ (𝑥 ∈ On → (𝜑 → [suc 𝑥 / 𝑥]𝜑)) |
tfindes.3 | ⊢ (Lim 𝑦 → (∀𝑥 ∈ 𝑦 𝜑 → [𝑦 / 𝑥]𝜑)) |
Ref | Expression |
---|---|
tfindes | ⊢ (𝑥 ∈ On → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsbcq 3590 | . 2 ⊢ (𝑦 = ∅ → ([𝑦 / 𝑥]𝜑 ↔ [∅ / 𝑥]𝜑)) | |
2 | dfsbcq 3590 | . 2 ⊢ (𝑦 = 𝑧 → ([𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑)) | |
3 | dfsbcq 3590 | . 2 ⊢ (𝑦 = suc 𝑧 → ([𝑦 / 𝑥]𝜑 ↔ [suc 𝑧 / 𝑥]𝜑)) | |
4 | sbceq2a 3600 | . 2 ⊢ (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝜑 ↔ 𝜑)) | |
5 | tfindes.1 | . 2 ⊢ [∅ / 𝑥]𝜑 | |
6 | nfv 1995 | . . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ On | |
7 | nfsbc1v 3608 | . . . . 5 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑 | |
8 | nfsbc1v 3608 | . . . . 5 ⊢ Ⅎ𝑥[suc 𝑧 / 𝑥]𝜑 | |
9 | 7, 8 | nfim 1977 | . . . 4 ⊢ Ⅎ𝑥([𝑧 / 𝑥]𝜑 → [suc 𝑧 / 𝑥]𝜑) |
10 | 6, 9 | nfim 1977 | . . 3 ⊢ Ⅎ𝑥(𝑧 ∈ On → ([𝑧 / 𝑥]𝜑 → [suc 𝑧 / 𝑥]𝜑)) |
11 | eleq1w 2833 | . . . 4 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ On ↔ 𝑧 ∈ On)) | |
12 | sbceq1a 3599 | . . . . 5 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑)) | |
13 | suceq 5934 | . . . . . 6 ⊢ (𝑥 = 𝑧 → suc 𝑥 = suc 𝑧) | |
14 | 13 | sbceq1d 3593 | . . . . 5 ⊢ (𝑥 = 𝑧 → ([suc 𝑥 / 𝑥]𝜑 ↔ [suc 𝑧 / 𝑥]𝜑)) |
15 | 12, 14 | imbi12d 333 | . . . 4 ⊢ (𝑥 = 𝑧 → ((𝜑 → [suc 𝑥 / 𝑥]𝜑) ↔ ([𝑧 / 𝑥]𝜑 → [suc 𝑧 / 𝑥]𝜑))) |
16 | 11, 15 | imbi12d 333 | . . 3 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ On → (𝜑 → [suc 𝑥 / 𝑥]𝜑)) ↔ (𝑧 ∈ On → ([𝑧 / 𝑥]𝜑 → [suc 𝑧 / 𝑥]𝜑)))) |
17 | tfindes.2 | . . 3 ⊢ (𝑥 ∈ On → (𝜑 → [suc 𝑥 / 𝑥]𝜑)) | |
18 | 10, 16, 17 | chvar 2424 | . 2 ⊢ (𝑧 ∈ On → ([𝑧 / 𝑥]𝜑 → [suc 𝑧 / 𝑥]𝜑)) |
19 | cbvralsv 3331 | . . . 4 ⊢ (∀𝑥 ∈ 𝑦 𝜑 ↔ ∀𝑧 ∈ 𝑦 [𝑧 / 𝑥]𝜑) | |
20 | sbsbc 3592 | . . . . 5 ⊢ ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑) | |
21 | 20 | ralbii 3129 | . . . 4 ⊢ (∀𝑧 ∈ 𝑦 [𝑧 / 𝑥]𝜑 ↔ ∀𝑧 ∈ 𝑦 [𝑧 / 𝑥]𝜑) |
22 | 19, 21 | bitri 264 | . . 3 ⊢ (∀𝑥 ∈ 𝑦 𝜑 ↔ ∀𝑧 ∈ 𝑦 [𝑧 / 𝑥]𝜑) |
23 | tfindes.3 | . . 3 ⊢ (Lim 𝑦 → (∀𝑥 ∈ 𝑦 𝜑 → [𝑦 / 𝑥]𝜑)) | |
24 | 22, 23 | syl5bir 233 | . 2 ⊢ (Lim 𝑦 → (∀𝑧 ∈ 𝑦 [𝑧 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜑)) |
25 | 1, 2, 3, 4, 5, 18, 24 | tfinds 7207 | 1 ⊢ (𝑥 ∈ On → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 [wsb 2049 ∈ wcel 2145 ∀wral 3061 [wsbc 3588 ∅c0 4064 Oncon0 5867 Lim wlim 5868 suc csuc 5869 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pr 5035 ax-un 7097 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 829 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3589 df-dif 3727 df-un 3729 df-in 3731 df-ss 3738 df-pss 3740 df-nul 4065 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-br 4788 df-opab 4848 df-tr 4888 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-we 5211 df-ord 5870 df-on 5871 df-lim 5872 df-suc 5873 |
This theorem is referenced by: tfinds2 7211 |
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