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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 3793 | . 2 ⊢ (𝑦 = ∅ → ([𝑦 / 𝑥]𝜑 ↔ [∅ / 𝑥]𝜑)) | |
2 | dfsbcq 3793 | . 2 ⊢ (𝑦 = 𝑧 → ([𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑)) | |
3 | dfsbcq 3793 | . 2 ⊢ (𝑦 = suc 𝑧 → ([𝑦 / 𝑥]𝜑 ↔ [suc 𝑧 / 𝑥]𝜑)) | |
4 | sbceq2a 3803 | . 2 ⊢ (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝜑 ↔ 𝜑)) | |
5 | tfindes.1 | . 2 ⊢ [∅ / 𝑥]𝜑 | |
6 | nfv 1912 | . . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ On | |
7 | nfsbc1v 3811 | . . . . 5 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑 | |
8 | nfsbc1v 3811 | . . . . 5 ⊢ Ⅎ𝑥[suc 𝑧 / 𝑥]𝜑 | |
9 | 7, 8 | nfim 1894 | . . . 4 ⊢ Ⅎ𝑥([𝑧 / 𝑥]𝜑 → [suc 𝑧 / 𝑥]𝜑) |
10 | 6, 9 | nfim 1894 | . . 3 ⊢ Ⅎ𝑥(𝑧 ∈ On → ([𝑧 / 𝑥]𝜑 → [suc 𝑧 / 𝑥]𝜑)) |
11 | eleq1w 2822 | . . . 4 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ On ↔ 𝑧 ∈ On)) | |
12 | sbceq1a 3802 | . . . . 5 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑)) | |
13 | suceq 6452 | . . . . . 6 ⊢ (𝑥 = 𝑧 → suc 𝑥 = suc 𝑧) | |
14 | 13 | sbceq1d 3796 | . . . . 5 ⊢ (𝑥 = 𝑧 → ([suc 𝑥 / 𝑥]𝜑 ↔ [suc 𝑧 / 𝑥]𝜑)) |
15 | 12, 14 | imbi12d 344 | . . . 4 ⊢ (𝑥 = 𝑧 → ((𝜑 → [suc 𝑥 / 𝑥]𝜑) ↔ ([𝑧 / 𝑥]𝜑 → [suc 𝑧 / 𝑥]𝜑))) |
16 | 11, 15 | imbi12d 344 | . . 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 | chvarfv 2238 | . 2 ⊢ (𝑧 ∈ On → ([𝑧 / 𝑥]𝜑 → [suc 𝑧 / 𝑥]𝜑)) |
19 | cbvralsvw 3315 | . . . 4 ⊢ (∀𝑥 ∈ 𝑦 𝜑 ↔ ∀𝑧 ∈ 𝑦 [𝑧 / 𝑥]𝜑) | |
20 | sbsbc 3795 | . . . . 5 ⊢ ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑) | |
21 | 20 | ralbii 3091 | . . . 4 ⊢ (∀𝑧 ∈ 𝑦 [𝑧 / 𝑥]𝜑 ↔ ∀𝑧 ∈ 𝑦 [𝑧 / 𝑥]𝜑) |
22 | 19, 21 | bitri 275 | . . 3 ⊢ (∀𝑥 ∈ 𝑦 𝜑 ↔ ∀𝑧 ∈ 𝑦 [𝑧 / 𝑥]𝜑) |
23 | tfindes.3 | . . 3 ⊢ (Lim 𝑦 → (∀𝑥 ∈ 𝑦 𝜑 → [𝑦 / 𝑥]𝜑)) | |
24 | 22, 23 | biimtrrid 243 | . 2 ⊢ (Lim 𝑦 → (∀𝑧 ∈ 𝑦 [𝑧 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜑)) |
25 | 1, 2, 3, 4, 5, 18, 24 | tfinds 7881 | 1 ⊢ (𝑥 ∈ On → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 [wsb 2062 ∈ wcel 2106 ∀wral 3059 [wsbc 3791 ∅c0 4339 Oncon0 6386 Lim wlim 6387 suc csuc 6388 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-tr 5266 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 |
This theorem is referenced by: tfinds2 7885 rdgssun 37361 |
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