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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 2731 | . 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 7895 | . 2 ⊢ (𝑥 ∈ ω → (∅ = ∅ → 𝜑)) |
10 | 1, 9 | mpi 20 | 1 ⊢ (𝑥 ∈ ω → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1540 ∈ wcel 2105 ∅c0 4322 suc csuc 6366 ωcom 7859 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-tr 5266 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-om 7860 |
This theorem is referenced by: findcard 9169 findcard2 9170 findcard2OLD 9290 pwfiOLD 9353 alephfplem3 10107 pwsdompw 10205 hsmexlem4 10430 |
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