Theorem nnindALT 9007

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 9006 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 9006	1 |- ( A e. NN -> ta )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   e. wcel 2167  (class class class)co 5922   1c1 7880   + caddc 7882   NNcn 8990

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-sep 4151  ax-cnex 7970  ax-resscn 7971  ax-1re 7973  ax-addrcl 7976

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-iota 5219  df-fv 5266  df-ov 5925  df-inn 8991

This theorem is referenced by: (None)