Theorem nnindALT 8931

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 8930 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 /allow". (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 8930	1 ⊢ (𝐴 ∈ ℕ → 𝜏)

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1353   ∈ wcel 2148  (class class class)co 5871  1c1 7808   + caddc 7810  ℕcn 8914

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-sep 4120  ax-cnex 7898  ax-resscn 7899  ax-1re 7901  ax-addrcl 7904

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4003  df-iota 5176  df-fv 5222  df-ov 5874  df-inn 8915

This theorem is referenced by: (None)