Theorem nn1suc 7122

Description: If a statement holds for 1 and also holds for a successor, it holds for all natural numbers. The first three hypotheses give us the substitution instances we need; the last two show that it holds for 1 and for a successor.

Hypotheses

Ref	Expression
nn1suc.1	|- (x = 1 -> (ph <-> ps))
nn1suc.3	|- (x = (y + 1) -> (ph <-> ch))
nn1suc.4	|- (x = A -> (ph <-> th))
nn1suc.5	|- ps
nn1suc.6	|- (y e. NN -> ch)

Assertion

Ref	Expression
nn1suc	|- (A e. NN -> th)

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

Proof of Theorem nn1suc

Step	Hyp	Ref	Expression
1		dfsbcq 2455	. . 3 |- (z = 1 -> ([z / x](A e. NN -> ph) <-> [1 / x](A e. NN -> ph)))
2		sbequ 1599	. . 3 |- (z = y -> ([z / x](A e. NN -> ph) <-> [y / x](A e. NN -> ph)))
3		dfsbcq 2455	. . 3 |- (z = (y + 1) -> ([z / x](A e. NN -> ph) <-> [(y + 1) / x](A e. NN -> ph)))
4		dfsbcq 2455	. . . . . . 7 |- (z = A -> ([z / x]ph <-> [A / x]ph))
5		elex 2302	. . . . . . . . . 10 |- (A e. NN -> E.x x = A)
6		ax-17 1317	. . . . . . . . . . . . 13 |- (z e. A -> A.x z e. A)
7	6	hbsbc1 2462	. . . . . . . . . . . 12 |- ((A e. NN -> [A / x]ph) -> A.x(A e. NN -> [A / x]ph))
8		ax-17 1317	. . . . . . . . . . . 12 |- ((A e. NN -> th) -> A.x(A e. NN -> th))
9	7, 8	hbbi 1357	. . . . . . . . . . 11 |- (((A e. NN -> [A / x]ph) <-> (A e. NN -> th)) -> A.x((A e. NN -> [A / x]ph) <-> (A e. NN -> th)))
10		sbceq1a 2456	. . . . . . . . . . . . 13 |- (x = A -> (ph <-> [A / x]ph))
11		nn1suc.4	. . . . . . . . . . . . 13 |- (x = A -> (ph <-> th))
12	10, 11	bitr3d 589	. . . . . . . . . . . 12 |- (x = A -> ([A / x]ph <-> th))
13	12	imbi2d 674	. . . . . . . . . . 11 |- (x = A -> ((A e. NN -> [A / x]ph) <-> (A e. NN -> th)))
14	9, 13	19.23ai 1412	. . . . . . . . . 10 |- (E.x x = A -> ((A e. NN -> [A / x]ph) <-> (A e. NN -> th)))
15	5, 14	syl 12	. . . . . . . . 9 |- (A e. NN -> ((A e. NN -> [A / x]ph) <-> (A e. NN -> th)))
16	15	pm5.74rd 648	. . . . . . . 8 |- (A e. NN -> (A e. NN -> ([A / x]ph <-> th)))
17	16	pm2.43i 78	. . . . . . 7 |- (A e. NN -> ([A / x]ph <-> th))
18	4, 17	sylan9bbr 600	. . . . . 6 |- ((A e. NN /\ z = A) -> ([z / x]ph <-> th))
19	18	expcom 403	. . . . 5 |- (z = A -> (A e. NN -> ([z / x]ph <-> th)))
20	19	pm5.74d 645	. . . 4 |- (z = A -> ((A e. NN -> [z / x]ph) <-> (A e. NN -> th)))
21		ax-17 1317	. . . . 5 |- (A e. NN -> A.x A e. NN)
22	21	sb19.21 1606	. . . 4 |- ([z / x](A e. NN -> ph) <-> (A e. NN -> [z / x]ph))
23	20, 22	syl5bb 591	. . 3 |- (z = A -> ([z / x](A e. NN -> ph) <-> (A e. NN -> th)))
24		1nn 7117	. . . . . . . 8 |- 1 e. NN
25	24	elisseti 2301	. . . . . . 7 |- 1 e. _V
26	25	isseti 2297	. . . . . 6 |- E.x x = 1
27	25	hbsbc1v 2464	. . . . . . 7 |- ([1 / x]ph -> A.x[1 / x]ph)
28		nn1suc.5	. . . . . . . . 9 |- ps
29		nn1suc.1	. . . . . . . . 9 |- (x = 1 -> (ph <-> ps))
30	28, 29	mpbiri 211	. . . . . . . 8 |- (x = 1 -> ph)
31		sbceq1a 2456	. . . . . . . 8 |- (x = 1 -> (ph <-> [1 / x]ph))
32	30, 31	mpbid 212	. . . . . . 7 |- (x = 1 -> [1 / x]ph)
33	27, 32	19.23ai 1412	. . . . . 6 |- (E.x x = 1 -> [1 / x]ph)
34	26, 33	ax-mp 7	. . . . 5 |- [1 / x]ph
35	34	a1i 8	. . . 4 |- (A e. NN -> [1 / x]ph)
36	21	sbc19.21g 2522	. . . . 5 |- (1 e. _V -> ([1 / x](A e. NN -> ph) <-> (A e. NN -> [1 / x]ph)))
37	25, 36	ax-mp 7	. . . 4 |- ([1 / x](A e. NN -> ph) <-> (A e. NN -> [1 / x]ph))
38	35, 37	mpbir 207	. . 3 |- [1 / x](A e. NN -> ph)
39		nn1suc.6	. . . . . . 7 |- (y e. NN -> ch)
40		oprex 4907	. . . . . . . . 9 |- (y + 1) e. _V
41	40	isseti 2297	. . . . . . . 8 |- E.x x = (y + 1)
42		ax-17 1317	. . . . . . . . . 10 |- (ch -> A.xch)
43	40	hbsbc1v 2464	. . . . . . . . . 10 |- ([(y + 1) / x]ph -> A.x[(y + 1) / x]ph)
44	42, 43	hbbi 1357	. . . . . . . . 9 |- ((ch <-> [(y + 1) / x]ph) -> A.x(ch <-> [(y + 1) / x]ph))
45		nn1suc.3	. . . . . . . . . 10 |- (x = (y + 1) -> (ph <-> ch))
46		sbceq1a 2456	. . . . . . . . . 10 |- (x = (y + 1) -> (ph <-> [(y + 1) / x]ph))
47	45, 46	bitr3d 589	. . . . . . . . 9 |- (x = (y + 1) -> (ch <-> [(y + 1) / x]ph))
48	44, 47	19.23ai 1412	. . . . . . . 8 |- (E.x x = (y + 1) -> (ch <-> [(y + 1) / x]ph))
49	41, 48	ax-mp 7	. . . . . . 7 |- (ch <-> [(y + 1) / x]ph)
50	39, 49	sylib 215	. . . . . 6 |- (y e. NN -> [(y + 1) / x]ph)
51	50	a1d 15	. . . . 5 |- (y e. NN -> (A e. NN -> [(y + 1) / x]ph))
52	21	sbc19.21g 2522	. . . . . 6 |- ((y + 1) e. _V -> ([(y + 1) / x](A e. NN -> ph) <-> (A e. NN -> [(y + 1) / x]ph)))
53	40, 52	ax-mp 7	. . . . 5 |- ([(y + 1) / x](A e. NN -> ph) <-> (A e. NN -> [(y + 1) / x]ph))
54	51, 53	sylibr 217	. . . 4 |- (y e. NN -> [(y + 1) / x](A e. NN -> ph))
55	54	a1d 15	. . 3 |- (y e. NN -> ([y / x](A e. NN -> ph) -> [(y + 1) / x](A e. NN -> ph)))
56	1, 2, 3, 23, 38, 55	nnind 7120	. 2 |- (A e. NN -> (A e. NN -> th))
57	56	pm2.43i 78	1 |- (A e. NN -> th)

Syntax hints:   -> wi 3   <-> wb 163   = wceq 1298   e. wcel 1300  E.wex 1326  [wsbc 1534  _Vcvv 2292  (class class class)co 4884  1c1 6387   + caddc 6389  NNcn 6449

This theorem is referenced by:  nnleltp1 7138  ruclem29 8807  opsqrlem6 11716

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-sub 6511  df-neg 6513  df-n 7108

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