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Theorem uzind4ALT 9384
 Description: Induction on the upper set of integers that starts at an integer 𝑀. The last four hypotheses give us the substitution instances we need; the first two are the basis and the induction step. Either uzind4 9383 or uzind4ALT 9384 may be used; see comment for nnind 8736. (Contributed by NM, 7-Sep-2005.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
uzind4ALT.5 (𝑀 ∈ ℤ → 𝜓)
uzind4ALT.6 (𝑘 ∈ (ℤ𝑀) → (𝜒𝜃))
uzind4ALT.1 (𝑗 = 𝑀 → (𝜑𝜓))
uzind4ALT.2 (𝑗 = 𝑘 → (𝜑𝜒))
uzind4ALT.3 (𝑗 = (𝑘 + 1) → (𝜑𝜃))
uzind4ALT.4 (𝑗 = 𝑁 → (𝜑𝜏))
Assertion
Ref Expression
uzind4ALT (𝑁 ∈ (ℤ𝑀) → 𝜏)
Distinct variable groups:   𝑗,𝑁   𝜓,𝑗   𝜒,𝑗   𝜃,𝑗   𝜏,𝑗   𝜑,𝑘   𝑗,𝑘,𝑀
Allowed substitution hints:   𝜑(𝑗)   𝜓(𝑘)   𝜒(𝑘)   𝜃(𝑘)   𝜏(𝑘)   𝑁(𝑘)

Proof of Theorem uzind4ALT
StepHypRef Expression
1 uzind4ALT.1 . 2 (𝑗 = 𝑀 → (𝜑𝜓))
2 uzind4ALT.2 . 2 (𝑗 = 𝑘 → (𝜑𝜒))
3 uzind4ALT.3 . 2 (𝑗 = (𝑘 + 1) → (𝜑𝜃))
4 uzind4ALT.4 . 2 (𝑗 = 𝑁 → (𝜑𝜏))
5 uzind4ALT.5 . 2 (𝑀 ∈ ℤ → 𝜓)
6 uzind4ALT.6 . 2 (𝑘 ∈ (ℤ𝑀) → (𝜒𝜃))
71, 2, 3, 4, 5, 6uzind4 9383 1 (𝑁 ∈ (ℤ𝑀) → 𝜏)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104   = wceq 1331   ∈ wcel 1480  ‘cfv 5123  (class class class)co 5774  1c1 7621   + caddc 7623  ℤcz 9054  ℤ≥cuz 9326 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-addcom 7720  ax-addass 7722  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-0id 7728  ax-rnegex 7729  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-ltadd 7736 This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-inn 8721  df-n0 8978  df-z 9055  df-uz 9327 This theorem is referenced by: (None)
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