Theorem nnindf 29565

Description: Principle of Mathematical Induction, using a bound-variable hypothesis instead of distinct variables. (Contributed by Thierry Arnoux, 6-May-2018.)

Hypotheses

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

Assertion

Ref	Expression
nnindf	|- ( A e. NN -> ta )

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

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

Proof of Theorem nnindf

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		1nn 11031	. . . . . 6 |- 1 e. NN
2		nnindf.5	. . . . . 6 |- ps
3		nnindf.1	. . . . . . 7 |- ( x = 1 -> ( ph <-> ps ) )
4	3	elrab 3363	. . . . . 6 |- ( 1 e. { x e. NN | ph } <-> ( 1 e. NN /\ ps ) )
5	1, 2, 4	mpbir2an 955	. . . . 5 |- 1 e. { x e. NN | ph }
6		elrabi 3359	. . . . . . . 8 |- ( y e. { x e. NN | ph } -> y e. NN )
7		peano2nn 11032	. . . . . . . . . . 11 |- ( y e. NN -> ( y + 1 ) e. NN )
8	7	a1d 25	. . . . . . . . . 10 |- ( y e. NN -> ( y e. NN -> ( y + 1 ) e. NN ) )
9		nnindf.6	. . . . . . . . . 10 |- ( y e. NN -> ( ch -> th ) )
10	8, 9	anim12d 586	. . . . . . . . 9 |- ( y e. NN -> ( ( y e. NN /\ ch ) -> ( ( y + 1 ) e. NN /\ th ) ) )
11		nnindf.2	. . . . . . . . . 10 |- ( x = y -> ( ph <-> ch ) )
12	11	elrab 3363	. . . . . . . . 9 |- ( y e. { x e. NN | ph } <-> ( y e. NN /\ ch ) )
13		nnindf.3	. . . . . . . . . 10 |- ( x = ( y + 1 ) -> ( ph <-> th ) )
14	13	elrab 3363	. . . . . . . . 9 |- ( ( y + 1 ) e. { x e. NN | ph } <-> ( ( y + 1 ) e. NN /\ th ) )
15	10, 12, 14	3imtr4g 285	. . . . . . . 8 |- ( y e. NN -> ( y e. { x e. NN | ph } -> ( y + 1 ) e. { x e. NN | ph } ) )
16	6, 15	mpcom 38	. . . . . . 7 |- ( y e. { x e. NN | ph } -> ( y + 1 ) e. { x e. NN | ph } )
17	16	rgen 2922	. . . . . 6 |- A. y e. { x e. NN | ph } ( y + 1 ) e. { x e. NN | ph }
18		nnindf.x	. . . . . . . 8 |- F/ y ph
19		nfcv 2764	. . . . . . . 8 |- F/_ y NN
20	18, 19	nfrab 3123	. . . . . . 7 |- F/_ y { x e. NN | ph }
21		nfcv 2764	. . . . . . 7 |- F/_ w { x e. NN | ph }
22		nfv 1843	. . . . . . 7 |- F/ w ( y + 1 ) e. { x e. NN | ph }
23	20	nfel2 2781	. . . . . . 7 |- F/ y ( w + 1 ) e. { x e. NN | ph }
24		oveq1 6657	. . . . . . . 8 |- ( y = w -> ( y + 1 ) = ( w + 1 ) )
25	24	eleq1d 2686	. . . . . . 7 |- ( y = w -> ( ( y + 1 ) e. { x e. NN | ph } <-> ( w + 1 ) e. { x e. NN | ph } ) )
26	20, 21, 22, 23, 25	cbvralf 3165	. . . . . 6 |- ( A. y e. { x e. NN | ph } ( y + 1 ) e. { x e. NN | ph } <-> A. w e. { x e. NN | ph } ( w + 1 ) e. { x e. NN | ph } )
27	17, 26	mpbi 220	. . . . 5 |- A. w e. { x e. NN | ph } ( w + 1 ) e. { x e. NN | ph }
28		peano5nni 11023	. . . . 5 |- ( ( 1 e. { x e. NN | ph } /\ A. w e. { x e. NN | ph } ( w + 1 ) e. { x e. NN | ph } ) -> NN C_ { x e. NN | ph } )
29	5, 27, 28	mp2an 708	. . . 4 |- NN C_ { x e. NN | ph }
30	29	sseli 3599	. . 3 |- ( A e. NN -> A e. { x e. NN | ph } )
31		nnindf.4	. . . 4 |- ( x = A -> ( ph <-> ta ) )
32	31	elrab 3363	. . 3 |- ( A e. { x e. NN | ph } <-> ( A e. NN /\ ta ) )
33	30, 32	sylib 208	. 2 |- ( A e. NN -> ( A e. NN /\ ta ) )
34	33	simprd 479	1 |- ( A e. NN -> ta )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   F/wnf 1708   e. wcel 1990   A.wral 2912   {crab 2916   C_ wss 3574  (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:  nn0min  29567