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Theorem nnindALT 12180
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 12179 has a different hypothesis order. It may be easier to use with the Metamath program 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_WITH 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 12179 1 (𝐴 ∈ ℕ → 𝜏)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1542  wcel 2107  (class class class)co 7361  1c1 11060   + caddc 11062  cn 12161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676  ax-1cn 11117
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-om 7807  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-nn 12162
This theorem is referenced by: (None)
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