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Theorem nnindALT 9953
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 9952 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 9952 1  |-  ( A  e.  NN  ->  ta )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649    e. wcel 1717  (class class class)co 6022   1c1 8926    + caddc 8928   NNcn 9934
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-1cn 8983
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-recs 6571  df-rdg 6606  df-nn 9935
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