Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2824 | . 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 7613 | . 2 ⊢ (𝑥 ∈ ω → (∅ = ∅ → 𝜑)) |
10 | 1, 9 | mpi 20 | 1 ⊢ (𝑥 ∈ ω → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1536 ∈ wcel 2113 ∅c0 4294 suc csuc 6196 ωcom 7583 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-tr 5176 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-om 7584 |
This theorem is referenced by: findcard 8760 findcard2 8761 pwfi 8822 alephfplem3 9535 pwsdompw 9629 hsmexlem4 9854 |
Copyright terms: Public domain | W3C validator |