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Theorem finds 4250
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.)
Hypotheses
Ref Expression
finds.1 (x = ∅ → (φψ))
finds.2 (x = y → (φχ))
finds.3 (x = suc y → (φθ))
finds.4 (x = A → (φτ))
finds.5 ψ
finds.6 (y 𝜔 → (χθ))
Assertion
Ref Expression
finds (A 𝜔 → τ)
Distinct variable groups:   x,y   x,A   ψ,x   χ,x   θ,x   τ,x   φ,y
Allowed substitution hints:   φ(x)   ψ(y)   χ(y)   θ(y)   τ(y)   A(y)

Proof of Theorem finds
StepHypRef Expression
1 finds.5 . . . . 5 ψ
2 0ex 3858 . . . . . 6 V
3 finds.1 . . . . . 6 (x = ∅ → (φψ))
42, 3elab 2664 . . . . 5 (∅ {xφ} ↔ ψ)
51, 4mpbir 134 . . . 4 {xφ}
6 finds.6 . . . . . 6 (y 𝜔 → (χθ))
7 vex 2538 . . . . . . 7 y V
8 finds.2 . . . . . . 7 (x = y → (φχ))
97, 8elab 2664 . . . . . 6 (y {xφ} ↔ χ)
107sucex 4175 . . . . . . 7 suc y V
11 finds.3 . . . . . . 7 (x = suc y → (φθ))
1210, 11elab 2664 . . . . . 6 (suc y {xφ} ↔ θ)
136, 9, 123imtr4g 194 . . . . 5 (y 𝜔 → (y {xφ} → suc y {xφ}))
1413rgen 2352 . . . 4 y 𝜔 (y {xφ} → suc y {xφ})
15 peano5 4248 . . . 4 ((∅ {xφ} y 𝜔 (y {xφ} → suc y {xφ})) → 𝜔 ⊆ {xφ})
165, 14, 15mp2an 404 . . 3 𝜔 ⊆ {xφ}
1716sseli 2918 . 2 (A 𝜔 → A {xφ})
18 finds.4 . . 3 (x = A → (φτ))
1918elabg 2665 . 2 (A 𝜔 → (A {xφ} ↔ τ))
2017, 19mpbid 135 1 (A 𝜔 → τ)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1228   wcel 1374  {cab 2008  wral 2284  wss 2894  c0 3201  suc csuc 4051  𝜔com 4240
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-nul 3857  ax-pow 3901  ax-pr 3918  ax-un 4120  ax-iinf 4238
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-pr 3357  df-uni 3555  df-int 3590  df-suc 4057  df-iom 4241
This theorem is referenced by:  findes  4253  nn0suc  4254  elnn  4255  ordom  4256  nndceq0  4266  0elnn  4267  nna0r  5972  nnm0r  5973  nnsucelsuc  5985
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