Theorem nnindALT 9271

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 9270 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 9270	1 |- ( A e. NN -> ta )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   e. wcel 2205  (class class class)co 6058   1c1 8144   + caddc 8146   NNcn 9254

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 2216  ax-sep 4233  ax-cnex 8234  ax-resscn 8235  ax-1re 8237  ax-addrcl 8240

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-iota 5317  df-fv 5365  df-ov 6061  df-inn 9255

This theorem is referenced by: (None)