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Mirrors > Home > MPE Home > Th. List > finds1 | Structured version Visualization version GIF version |
Description: Principle of Finite Induction (inference schema), using implicit substitutions. The first three hypotheses establish the substitutions we need. The last two are the basis and the induction step. Theorem Schema 22 of [Suppes] p. 136. (Contributed by NM, 22-Mar-2006.) |
Ref | Expression |
---|---|
finds1.1 | ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓)) |
finds1.2 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) |
finds1.3 | ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃)) |
finds1.4 | ⊢ 𝜓 |
finds1.5 | ⊢ (𝑦 ∈ ω → (𝜒 → 𝜃)) |
Ref | Expression |
---|---|
finds1 | ⊢ (𝑥 ∈ ω → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2610 | . 2 ⊢ ∅ = ∅ | |
2 | finds1.1 | . . 3 ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓)) | |
3 | finds1.2 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) | |
4 | finds1.3 | . . 3 ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃)) | |
5 | finds1.4 | . . . 4 ⊢ 𝜓 | |
6 | 5 | a1i 11 | . . 3 ⊢ (∅ = ∅ → 𝜓) |
7 | finds1.5 | . . . 4 ⊢ (𝑦 ∈ ω → (𝜒 → 𝜃)) | |
8 | 7 | a1d 25 | . . 3 ⊢ (𝑦 ∈ ω → (∅ = ∅ → (𝜒 → 𝜃))) |
9 | 2, 3, 4, 6, 8 | finds2 6986 | . 2 ⊢ (𝑥 ∈ ω → (∅ = ∅ → 𝜑)) |
10 | 1, 9 | mpi 20 | 1 ⊢ (𝑥 ∈ ω → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 ∅c0 3874 suc csuc 5642 ωcom 6957 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-tr 4681 df-eprel 4949 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-om 6958 |
This theorem is referenced by: findcard 8084 findcard2 8085 pwfi 8144 alephfplem3 8812 pwsdompw 8909 hsmexlem4 9134 |
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