Theorem nnindALT 9254

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 9253 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

Step	Hyp	Ref	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 ) )
7	1, 2, 3, 4, 5, 6	nnind 9253	1 |- ( A e. NN -> ta )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   e. wcel 2203  (class class class)co 6050   1c1 8128   + caddc 8130   NNcn 9237

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214  ax-sep 4228  ax-cnex 8218  ax-resscn 8219  ax-1re 8221  ax-addrcl 8224

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-iota 5312  df-fv 5360  df-ov 6053  df-inn 9238

This theorem is referenced by: (None)