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 → (φ ↔ ψ))
nnind.2	⊢ (x = y → (φ ↔ χ))
nnind.3	⊢ (x = (y + 1) → (φ ↔ θ))
nnind.4	⊢ (x = A → (φ ↔ τ))
nnind.5	⊢ ψ
nnind.6	⊢ (y ∈ ℕ → (χ → θ))

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

Step	Hyp	Ref	Expression
1		1nn 7666	. . . . . 6 ⊢ 1 ∈ ℕ
2		nnind.5	. . . . . 6 ⊢ ψ
3		nnind.1	. . . . . . 7 ⊢ (x = 1 → (φ ↔ ψ))
4	3	elrab 2692	. . . . . 6 ⊢ (1 ∈ {x ∈ ℕ ∣ φ} ↔ (1 ∈ ℕ ∧ ψ))
5	1, 2, 4	mpbir2an 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)
8	7	a1d 22	. . . . . . . . 9 ⊢ (y ∈ ℕ → (y ∈ ℕ → (y + 1) ∈ ℕ))
9		nnind.6	. . . . . . . . 9 ⊢ (y ∈ ℕ → (χ → θ))
10	8, 9	anim12d 318	. . . . . . . 8 ⊢ (y ∈ ℕ → ((y ∈ ℕ ∧ χ) → ((y + 1) ∈ ℕ ∧ θ)))
11		nnind.2	. . . . . . . . 9 ⊢ (x = y → (φ ↔ χ))
12	11	elrab 2692	. . . . . . . 8 ⊢ (y ∈ {x ∈ ℕ ∣ φ} ↔ (y ∈ ℕ ∧ χ))
13		nnind.3	. . . . . . . . 9 ⊢ (x = (y + 1) → (φ ↔ θ))
14	13	elrab 2692	. . . . . . . 8 ⊢ ((y + 1) ∈ {x ∈ ℕ ∣ φ} ↔ ((y + 1) ∈ ℕ ∧ θ))
15	10, 12, 14	3imtr4g 194	. . . . . . 7 ⊢ (y ∈ ℕ → (y ∈ {x ∈ ℕ ∣ φ} → (y + 1) ∈ {x ∈ ℕ ∣ φ}))
16	6, 15	mpcom 32	. . . . . 6 ⊢ (y ∈ {x ∈ ℕ ∣ φ} → (y + 1) ∈ {x ∈ ℕ ∣ φ})
17	16	rgen 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})
19	5, 17, 18	mp2an 402	. . . 4 ⊢ ℕ ⊆ {x ∈ ℕ ∣ φ}
20	19	sseli 2935	. . 3 ⊢ (A ∈ ℕ → A ∈ {x ∈ ℕ ∣ φ})
21		nnind.4	. . . 4 ⊢ (x = A → (φ ↔ τ))
22	21	elrab 2692	. . 3 ⊢ (A ∈ {x ∈ ℕ ∣ φ} ↔ (A ∈ ℕ ∧ τ))
23	20, 22	sylib 127	. 2 ⊢ (A ∈ ℕ → (A ∈ ℕ ∧ τ))
24	23	simprd 107	1 ⊢ (A ∈ ℕ → τ)

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  ℕcn 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