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Theorem nnindALT 8868
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 8867 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  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 8867 1  |-  ( A  e.  NN  ->  ta )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1342    e. wcel 2135  (class class class)co 5839   1c1 7748    + caddc 7750   NNcn 8851
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-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146  ax-sep 4097  ax-cnex 7838  ax-resscn 7839  ax-1re 7841  ax-addrcl 7844
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-rab 2451  df-v 2726  df-un 3118  df-in 3120  df-ss 3127  df-sn 3579  df-pr 3580  df-op 3582  df-uni 3787  df-int 3822  df-br 3980  df-iota 5150  df-fv 5193  df-ov 5842  df-inn 8852
This theorem is referenced by: (None)
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