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Theorem nnindALT 10009
Description: Principle of Mathematical Induction (inference schema). The last four hypotheses give us the substitution instances we need; the first two are the induction hypothesis and the basis.

This ALT version of nnind 10008 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.)

Hypotheses
Ref Expression
nnindALT.6  |-  ( y  e.  NN  ->  ( ch  ->  th ) )
nnindALT.5  |-  ps
nnindALT.1  |-  ( x  =  1  ->  ( ph 
<->  ps ) )
nnindALT.2  |-  ( x  =  y  ->  ( ph 
<->  ch ) )
nnindALT.3  |-  ( x  =  ( y  +  1 )  ->  ( ph 
<->  th ) )
nnindALT.4  |-  ( x  =  A  ->  ( ph 
<->  ta ) )
Assertion
Ref Expression
nnindALT  |-  ( A  e.  NN  ->  ta )
Distinct variable groups:    x, y    x, A    ps, x    ch, x    th, x    ta, x    ph, y
Allowed substitution hints:    ph( x)    ps( y)    ch( y)    th( y)    ta( y)    A( y)

Proof of Theorem nnindALT
StepHypRef Expression
1 nnindALT.1 . 2  |-  ( x  =  1  ->  ( ph 
<->  ps ) )
2 nnindALT.2 . 2  |-  ( x  =  y  ->  ( ph 
<->  ch ) )
3 nnindALT.3 . 2  |-  ( x  =  ( y  +  1 )  ->  ( ph 
<->  th ) )
4 nnindALT.4 . 2  |-  ( x  =  A  ->  ( ph 
<->  ta ) )
5 nnindALT.5 . 2  |-  ps
6 nnindALT.6 . 2  |-  ( y  e.  NN  ->  ( ch  ->  th ) )
71, 2, 3, 4, 5, 6nnind 10008 1  |-  ( A  e.  NN  ->  ta )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1652    e. wcel 1725  (class class class)co 6073   1c1 8981    + caddc 8983   NNcn 9990
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-1cn 9038
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-recs 6625  df-rdg 6660  df-nn 9991
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