# Metamath Proof Explorer

## Theorem finds

Description: Principle of Finite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last two are the basis and the induction step. Theorem Schema 22 of Suppes p. 136. This is Metamath 100 proof #74. (Contributed by NM, 14-Apr-1995)

Ref Expression
Hypotheses finds.1 ${⊢}{x}=\varnothing \to \left({\phi }↔{\psi }\right)$
finds.2 ${⊢}{x}={y}\to \left({\phi }↔{\chi }\right)$
finds.3 ${⊢}{x}=\mathrm{suc}{y}\to \left({\phi }↔{\theta }\right)$
finds.4 ${⊢}{x}={A}\to \left({\phi }↔{\tau }\right)$
finds.5 ${⊢}{\psi }$
finds.6 ${⊢}{y}\in \mathrm{\omega }\to \left({\chi }\to {\theta }\right)$
Assertion finds ${⊢}{A}\in \mathrm{\omega }\to {\tau }$

### Proof

Step Hyp Ref Expression
1 finds.1 ${⊢}{x}=\varnothing \to \left({\phi }↔{\psi }\right)$
2 finds.2 ${⊢}{x}={y}\to \left({\phi }↔{\chi }\right)$
3 finds.3 ${⊢}{x}=\mathrm{suc}{y}\to \left({\phi }↔{\theta }\right)$
4 finds.4 ${⊢}{x}={A}\to \left({\phi }↔{\tau }\right)$
5 finds.5 ${⊢}{\psi }$
6 finds.6 ${⊢}{y}\in \mathrm{\omega }\to \left({\chi }\to {\theta }\right)$
7 0ex ${⊢}\varnothing \in \mathrm{V}$
8 7 1 elab ${⊢}\varnothing \in \left\{{x}|{\phi }\right\}↔{\psi }$
9 5 8 mpbir ${⊢}\varnothing \in \left\{{x}|{\phi }\right\}$
10 vex ${⊢}{y}\in \mathrm{V}$
11 10 2 elab ${⊢}{y}\in \left\{{x}|{\phi }\right\}↔{\chi }$
12 10 sucex ${⊢}\mathrm{suc}{y}\in \mathrm{V}$
13 12 3 elab ${⊢}\mathrm{suc}{y}\in \left\{{x}|{\phi }\right\}↔{\theta }$
14 6 11 13 3imtr4g ${⊢}{y}\in \mathrm{\omega }\to \left({y}\in \left\{{x}|{\phi }\right\}\to \mathrm{suc}{y}\in \left\{{x}|{\phi }\right\}\right)$
15 14 rgen ${⊢}\forall {y}\in \mathrm{\omega }\phantom{\rule{.4em}{0ex}}\left({y}\in \left\{{x}|{\phi }\right\}\to \mathrm{suc}{y}\in \left\{{x}|{\phi }\right\}\right)$
16 peano5 ${⊢}\left(\varnothing \in \left\{{x}|{\phi }\right\}\wedge \forall {y}\in \mathrm{\omega }\phantom{\rule{.4em}{0ex}}\left({y}\in \left\{{x}|{\phi }\right\}\to \mathrm{suc}{y}\in \left\{{x}|{\phi }\right\}\right)\right)\to \mathrm{\omega }\subseteq \left\{{x}|{\phi }\right\}$
17 9 15 16 mp2an ${⊢}\mathrm{\omega }\subseteq \left\{{x}|{\phi }\right\}$
18 17 sseli ${⊢}{A}\in \mathrm{\omega }\to {A}\in \left\{{x}|{\phi }\right\}$
19 4 elabg ${⊢}{A}\in \mathrm{\omega }\to \left({A}\in \left\{{x}|{\phi }\right\}↔{\tau }\right)$
20 18 19 mpbid ${⊢}{A}\in \mathrm{\omega }\to {\tau }$