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Theorem nnind 7671
Description: Principle of Mathematical Induction (inference schema). The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. See nnaddcl 7675 for an example of its use. This is an alternative for Metamath 100 proof #74. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 16-Jun-2013.)
Hypotheses
Ref Expression
nnind.1 (x = 1 → (φψ))
nnind.2 (x = y → (φχ))
nnind.3 (x = (y + 1) → (φθ))
nnind.4 (x = A → (φτ))
nnind.5 ψ
nnind.6 (y ℕ → (χθ))
Assertion
Ref Expression
nnind (A ℕ → 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 nnind
StepHypRef Expression
1 1nn 7666 . . . . . 6 1
2 nnind.5 . . . . . 6 ψ
3 nnind.1 . . . . . . 7 (x = 1 → (φψ))
43elrab 2692 . . . . . 6 (1 {x ℕ ∣ φ} ↔ (1 ψ))
51, 2, 4mpbir2an 848 . . . . 5 1 {x ℕ ∣ φ}
6 elrabi 2689 . . . . . . 7 (y {x ℕ ∣ φ} → y ℕ)
7 peano2nn 7667 . . . . . . . . . 10 (y ℕ → (y + 1) ℕ)
87a1d 22 . . . . . . . . 9 (y ℕ → (y ℕ → (y + 1) ℕ))
9 nnind.6 . . . . . . . . 9 (y ℕ → (χθ))
108, 9anim12d 318 . . . . . . . 8 (y ℕ → ((y χ) → ((y + 1) θ)))
11 nnind.2 . . . . . . . . 9 (x = y → (φχ))
1211elrab 2692 . . . . . . . 8 (y {x ℕ ∣ φ} ↔ (y χ))
13 nnind.3 . . . . . . . . 9 (x = (y + 1) → (φθ))
1413elrab 2692 . . . . . . . 8 ((y + 1) {x ℕ ∣ φ} ↔ ((y + 1) θ))
1510, 12, 143imtr4g 194 . . . . . . 7 (y ℕ → (y {x ℕ ∣ φ} → (y + 1) {x ℕ ∣ φ}))
166, 15mpcom 32 . . . . . 6 (y {x ℕ ∣ φ} → (y + 1) {x ℕ ∣ φ})
1716rgen 2368 . . . . 5 y {x ℕ ∣ φ} (y + 1) {x ℕ ∣ φ}
18 peano5nni 7658 . . . . 5 ((1 {x ℕ ∣ φ} y {x ℕ ∣ φ} (y + 1) {x ℕ ∣ φ}) → ℕ ⊆ {x ℕ ∣ φ})
195, 17, 18mp2an 402 . . . 4 ℕ ⊆ {x ℕ ∣ φ}
2019sseli 2935 . . 3 (A ℕ → A {x ℕ ∣ φ})
21 nnind.4 . . . 4 (x = A → (φτ))
2221elrab 2692 . . 3 (A {x ℕ ∣ φ} ↔ (A τ))
2320, 22sylib 127 . 2 (A ℕ → (A τ))
2423simprd 107 1 (A ℕ → τ)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  wral 2300  {crab 2304  wss 2911  (class class class)co 5455  1c1 6672   + caddc 6674  cn 7655
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-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-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-cnex 6734  ax-resscn 6735  ax-1re 6737  ax-addrcl 6740
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-rab 2309  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-br 3756  df-iota 4810  df-fv 4853  df-ov 5458  df-inn 7656
This theorem is referenced by:  nnindALT  7672  nn1m1nn  7673  nnaddcl  7675  nnmulcl  7676  nnge1  7678  nn1gt1  7688  nnsub  7693  zaddcllempos  8018  zaddcllemneg  8020  nneoor  8076  peano5uzti  8082  nn0ind-raph  8091  indstr  8272  expivallem  8870  expcllem  8880  expap0  8899
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