Theorem nnindd 29566

Description: Principle of Mathematical Induction (inference schema) on integers, a deduction version. (Contributed by Thierry Arnoux, 19-Jul-2020.)

Hypotheses

Ref	Expression
nnindd.1	|- ( x = 1 -> ( ps <-> ch ) )
nnindd.2	|- ( x = y -> ( ps <-> th ) )
nnindd.3	|- ( x = ( y + 1 ) -> ( ps <-> ta ) )
nnindd.4	|- ( x = A -> ( ps <-> et ) )
nnindd.5	|- ( ph -> ch )
nnindd.6	|- ( ( ( ph /\ y e. NN ) /\ th ) -> ta )

Assertion

Ref	Expression
nnindd	|- ( ( ph /\ A e. NN ) -> et )

Distinct variable groups:   x, A   x, y, ph   ps, y   ch, x   et, x   th, x   ta, x

Allowed substitution hints:   ps( x)   ch( y)   th( y)   ta( y)   et( y)   A( y)

Proof of Theorem nnindd

Step	Hyp	Ref	Expression
1		nnindd.1	. . . 4 |- ( x = 1 -> ( ps <-> ch ) )
2	1	imbi2d 330	. . 3 |- ( x = 1 -> ( ( ph -> ps ) <-> ( ph -> ch ) ) )
3		nnindd.2	. . . 4 |- ( x = y -> ( ps <-> th ) )
4	3	imbi2d 330	. . 3 |- ( x = y -> ( ( ph -> ps ) <-> ( ph -> th ) ) )
5		nnindd.3	. . . 4 |- ( x = ( y + 1 ) -> ( ps <-> ta ) )
6	5	imbi2d 330	. . 3 |- ( x = ( y + 1 ) -> ( ( ph -> ps ) <-> ( ph -> ta ) ) )
7		nnindd.4	. . . 4 |- ( x = A -> ( ps <-> et ) )
8	7	imbi2d 330	. . 3 |- ( x = A -> ( ( ph -> ps ) <-> ( ph -> et ) ) )
9		nnindd.5	. . 3 |- ( ph -> ch )
10		nnindd.6	. . . . . 6 |- ( ( ( ph /\ y e. NN ) /\ th ) -> ta )
11	10	ex 450	. . . . 5 |- ( ( ph /\ y e. NN ) -> ( th -> ta ) )
12	11	expcom 451	. . . 4 |- ( y e. NN -> ( ph -> ( th -> ta ) ) )
13	12	a2d 29	. . 3 |- ( y e. NN -> ( ( ph -> th ) -> ( ph -> ta ) ) )
14	2, 4, 6, 8, 9, 13	nnind 11038	. 2 |- ( A e. NN -> ( ph -> et ) )
15	14	impcom 446	1 |- ( ( ph /\ A e. NN ) -> et )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990  (class class class)co 6650   1c1 9937   + caddc 9939   NNcn 11020

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-1cn 9994

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-nn 11021

This theorem is referenced by:  fzto1st  29853  psgnfzto1st  29855  fiunelros  30237