Theorem nnindALT 12191

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 12190 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 12190	1 ⊢ (𝐴 ∈ ℕ → 𝜏)

Syntax hints:   → wi 4   ↔ wb 207   = wceq 1547   ∈ wcel 2119  (class class class)co 7363  1c1 11037   + caddc 11039  ℕcn 12172

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369  ax-un 7685  ax-1cn 11094

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7366  df-om 7814  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-nn 12173

This theorem is referenced by: (None)