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| 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 2737 | . 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 7920 | . 2 ⊢ (𝑥 ∈ ω → (∅ = ∅ → 𝜑)) | 
| 10 | 1, 9 | mpi 20 | 1 ⊢ (𝑥 ∈ ω → 𝜑) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 ∅c0 4333 suc csuc 6386 ωcom 7887 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-om 7888 | 
| This theorem is referenced by: findcard 9203 findcard2 9204 alephfplem3 10146 pwsdompw 10243 hsmexlem4 10469 | 
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