Theorem nnindALT 9162

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 9161 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 9161	1 |- ( A e. NN -> ta )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1397   e. wcel 2201  (class class class)co 6020   1c1 8035   + caddc 8037   NNcn 9145

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2212  ax-sep 4206  ax-cnex 8125  ax-resscn 8126  ax-1re 8128  ax-addrcl 8131

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-rab 2518  df-v 2803  df-un 3203  df-in 3205  df-ss 3212  df-sn 3674  df-pr 3675  df-op 3677  df-uni 3893  df-int 3928  df-br 4088  df-iota 5285  df-fv 5333  df-ov 6023  df-inn 9146

This theorem is referenced by: (None)