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Theorem finds 4265
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 3874 . . . . . 6 V
3 finds.1 . . . . . 6 (x = ∅ → (φψ))
42, 3elab 2681 . . . . 5 (∅ {xφ} ↔ ψ)
51, 4mpbir 134 . . . 4 {xφ}
6 finds.6 . . . . . 6 (y 𝜔 → (χθ))
7 vex 2554 . . . . . . 7 y V
8 finds.2 . . . . . . 7 (x = y → (φχ))
97, 8elab 2681 . . . . . 6 (y {xφ} ↔ χ)
107sucex 4190 . . . . . . 7 suc y V
11 finds.3 . . . . . . 7 (x = suc y → (φθ))
1210, 11elab 2681 . . . . . 6 (suc y {xφ} ↔ θ)
136, 9, 123imtr4g 194 . . . . 5 (y 𝜔 → (y {xφ} → suc y {xφ}))
1413rgen 2368 . . . 4 y 𝜔 (y {xφ} → suc y {xφ})
15 peano5 4263 . . . 4 ((∅ {xφ} y 𝜔 (y {xφ} → suc y {xφ})) → 𝜔 ⊆ {xφ})
165, 14, 15mp2an 402 . . 3 𝜔 ⊆ {xφ}
1716sseli 2935 . 2 (A 𝜔 → A {xφ})
18 finds.4 . . 3 (x = A → (φτ))
1918elabg 2682 . 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 1242   wcel 1390  {cab 2023  wral 2300  wss 2911  c0 3218  suc csuc 4067  𝜔com 4255
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 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3865  ax-nul 3873  ax-pow 3917  ax-pr 3934  ax-un 4135  ax-iinf 4253
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3352  df-sn 3372  df-pr 3373  df-uni 3571  df-int 3606  df-suc 4073  df-iom 4256
This theorem is referenced by:  findes  4268  nn0suc  4269  elnn  4270  ordom  4271  nndceq0  4281  0elnn  4282  nna0r  5989  nnm0r  5990  nnsucelsuc  6002  frec2uzltd  8816
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