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Theorem nnindALT 10916
 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 10915 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 /maygrow";. (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 10915 1 (𝐴 ∈ ℕ → 𝜏)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977  (class class class)co 6549  1c1 9816   + caddc 9818  ℕcn 10897 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-1cn 9873 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-nn 10898 This theorem is referenced by: (None)
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