Theorem nnindALT 9053

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 9052 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 9052	1 |- ( A e. NN -> ta )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1373   e. wcel 2176  (class class class)co 5944   1c1 7926   + caddc 7928   NNcn 9036

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187  ax-sep 4162  ax-cnex 8016  ax-resscn 8017  ax-1re 8019  ax-addrcl 8022

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4045  df-iota 5232  df-fv 5279  df-ov 5947  df-inn 9037

This theorem is referenced by: (None)