Theorem nnindALT 8730

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 8729 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 8729	1 ⊢ (𝐴 ∈ ℕ → 𝜏)

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1331   ∈ wcel 1480  (class class class)co 5767  1c1 7614   + caddc 7616  ℕcn 8713

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-cnex 7704  ax-resscn 7705  ax-1re 7707  ax-addrcl 7710

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-br 3925  df-iota 5083  df-fv 5126  df-ov 5770  df-inn 8714

This theorem is referenced by: (None)