ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nnindALT GIF version

Theorem nnindALT 8694
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 8693 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 → (𝜒𝜃))
nnindALT.5 𝜓
nnindALT.1 (𝑥 = 1 → (𝜑𝜓))
nnindALT.2 (𝑥 = 𝑦 → (𝜑𝜒))
nnindALT.3 (𝑥 = (𝑦 + 1) → (𝜑𝜃))
nnindALT.4 (𝑥 = 𝐴 → (𝜑𝜏))
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
StepHypRef Expression
1 nnindALT.1 . 2 (𝑥 = 1 → (𝜑𝜓))
2 nnindALT.2 . 2 (𝑥 = 𝑦 → (𝜑𝜒))
3 nnindALT.3 . 2 (𝑥 = (𝑦 + 1) → (𝜑𝜃))
4 nnindALT.4 . 2 (𝑥 = 𝐴 → (𝜑𝜏))
5 nnindALT.5 . 2 𝜓
6 nnindALT.6 . 2 (𝑦 ∈ ℕ → (𝜒𝜃))
71, 2, 3, 4, 5, 6nnind 8693 1 (𝐴 ∈ ℕ → 𝜏)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1314  wcel 1463  (class class class)co 5740  1c1 7585   + caddc 7587  cn 8677
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-cnex 7675  ax-resscn 7676  ax-1re 7678  ax-addrcl 7681
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-br 3898  df-iota 5056  df-fv 5099  df-ov 5743  df-inn 8678
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator