Theorem nnindALT 8874

Description: Principle of Mathematical Induction (inference schema). The last four hypotheses give us the substitution instances we need; the first two are the induction step and the basis.

This ALT version of nnind 8873 has a different hypothesis order. It may be easier to use with the metamath program's Proof Assistant, because "MM-PA> assign last" will be applied to the substitution instances first. We may eventually use this one as the official version. You may use either version. After the proof is complete, the ALT version can be changed to the non-ALT version with "MM-PA> minimize nnind /allow". (Contributed by NM, 7-Dec-2005.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
nnindALT	⊢ (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 nnindALT

Step	Hyp	Ref	Expression
1		nnindALT.1	. 2 ⊢ (𝑥 = 1 → (𝜑 ↔ 𝜓))
2		nnindALT.2	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
3		nnindALT.3	. 2 ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜃))
4		nnindALT.4	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))
5		nnindALT.5	. 2 ⊢ 𝜓
6		nnindALT.6	. 2 ⊢ (𝑦 ∈ ℕ → (𝜒 → 𝜃))
7	1, 2, 3, 4, 5, 6	nnind 8873	1 ⊢ (𝐴 ∈ ℕ → 𝜏)

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1343   ∈ wcel 2136  (class class class)co 5842  1c1 7754   + caddc 7756  ℕcn 8857

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-sep 4100  ax-cnex 7844  ax-resscn 7845  ax-1re 7847  ax-addrcl 7850

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-iota 5153  df-fv 5196  df-ov 5845  df-inn 8858

This theorem is referenced by: (None)