Theorem nnindALT 12312

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 12311 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 / MAY_GROW";. (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 12311	1 ⊢ (𝐴 ∈ ℕ → 𝜏)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1537   ∈ wcel 2108  (class class class)co 7448  1c1 11185   + caddc 11187  ℕcn 12293

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770  ax-1cn 11242

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-om 7904  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-nn 12294

This theorem is referenced by: (None)