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Theorem nn0indALT 11349
 Description: Principle of Mathematical Induction (inference schema) on nonnegative integers. The last four hypotheses give us the substitution instances we need; the first two are the basis and the induction step. Either nn0ind 11348 or nn0indALT 11349 may be used; see comment for nnind 10915. (Contributed by NM, 28-Nov-2005.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
nn0indALT.6 (𝑦 ∈ ℕ0 → (𝜒𝜃))
nn0indALT.5 𝜓
nn0indALT.1 (𝑥 = 0 → (𝜑𝜓))
nn0indALT.2 (𝑥 = 𝑦 → (𝜑𝜒))
nn0indALT.3 (𝑥 = (𝑦 + 1) → (𝜑𝜃))
nn0indALT.4 (𝑥 = 𝐴 → (𝜑𝜏))
Assertion
Ref Expression
nn0indALT (𝐴 ∈ ℕ0𝜏)
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 nn0indALT
StepHypRef Expression
1 nn0indALT.1 . 2 (𝑥 = 0 → (𝜑𝜓))
2 nn0indALT.2 . 2 (𝑥 = 𝑦 → (𝜑𝜒))
3 nn0indALT.3 . 2 (𝑥 = (𝑦 + 1) → (𝜑𝜃))
4 nn0indALT.4 . 2 (𝑥 = 𝐴 → (𝜑𝜏))
5 nn0indALT.5 . 2 𝜓
6 nn0indALT.6 . 2 (𝑦 ∈ ℕ0 → (𝜒𝜃))
71, 2, 3, 4, 5, 6nn0ind 11348 1 (𝐴 ∈ ℕ0𝜏)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818  ℕ0cn0 11169 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-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 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-nel 2783  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-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255 This theorem is referenced by:  uzaddcl  11620  faclbnd4lem4  12945  ipasslem1  27070
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