Theorem nn1suc 5887

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 1933	. . 3 |- (z = 1 -> ([z / x](A e. NN -> ph) <-> [1 / x](A e. NN -> ph)))
2		sbequ 1224	. . 3 |- (z = y -> ([z / x](A e. NN -> ph) <-> [y / x](A e. NN -> ph)))
3		dfsbcq 1933	. . 3 |- (z = (y + 1) -> ([z / x](A e. NN -> ph) <-> [(y + 1) / x](A e. NN -> ph)))
4		dfsbcq 1933	. . . . . . 7 |- (z = A -> ([z / x]ph <-> [A / x]ph))
5		elex 1810	. . . . . . . . . 10 |- (A e. NN -> E.x x = A)
6		ax-17 968	. . . . . . . . . . . . 13 |- (z e. A -> A.x z e. A)
7	6	hbsbc1 1939	. . . . . . . . . . . 12 |- ((A e. NN -> [A / x]ph) -> A.x(A e. NN -> [A / x]ph))
8		ax-17 968	. . . . . . . . . . . 12 |- ((A e. NN -> th) -> A.x(A e. NN -> th))
9	7, 8	hbbi 1007	. . . . . . . . . . 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 1934	. . . . . . . . . . . . 13 |- (x = A -> (ph <-> [A / x]ph))
11		nn1suc.4	. . . . . . . . . . . . 13 |- (x = A -> (ph <-> th))
12	10, 11	bitr3d 528	. . . . . . . . . . . 12 |- (x = A -> ([A / x]ph <-> th))
13	12	imbi2d 610	. . . . . . . . . . 11 |- (x = A -> ((A e. NN -> [A / x]ph) <-> (A e. NN -> th)))
14	9, 13	19.23ai 1060	. . . . . . . . . 10 |- (E.x x = A -> ((A e. NN -> [A / x]ph) <-> (A e. NN -> th)))
15	5, 14	syl 10	. . . . . . . . 9 |- (A e. NN -> ((A e. NN -> [A / x]ph) <-> (A e. NN -> th)))
16	15	pm5.74rd 586	. . . . . . . 8 |- (A e. NN -> (A e. NN -> ([A / x]ph <-> th)))
17	16	pm2.43i 64	. . . . . . 7 |- (A e. NN -> ([A / x]ph <-> th))
18	4, 17	sylan9bbr 539	. . . . . 6 |- ((A e. NN /\ z = A) -> ([z / x]ph <-> th))
19	18	expcom 374	. . . . 5 |- (z = A -> (A e. NN -> ([z / x]ph <-> th)))
20	19	pm5.74d 583	. . . 4 |- (z = A -> ((A e. NN -> [z / x]ph) <-> (A e. NN -> th)))
21		ax-17 968	. . . . 5 |- (A e. NN -> A.x A e. NN)
22	21	sb19.21 1231	. . . 4 |- ([z / x](A e. NN -> ph) <-> (A e. NN -> [z / x]ph))
23	20, 22	syl5bb 530	. . 3 |- (z = A -> ([z / x](A e. NN -> ph) <-> (A e. NN -> th)))
24		1nn 5882	. . . . . . . 8 |- 1 e. NN
25	24	elisseti 1809	. . . . . . 7 |- 1 e. V
26	25	isseti 1806	. . . . . 6 |- E.x x = 1
27	25	hbsbc1v 1940	. . . . . . 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 194	. . . . . . . 8 |- (x = 1 -> ph)
31		sbceq1a 1934	. . . . . . . 8 |- (x = 1 -> (ph <-> [1 / x]ph))
32	30, 31	mpbid 195	. . . . . . 7 |- (x = 1 -> [1 / x]ph)
33	27, 32	19.23ai 1060	. . . . . 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 1977	. . . . 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 190	. . 3 |- [1 / x](A e. NN -> ph)
39		nn1suc.6	. . . . . . 7 |- (y e. NN -> ch)
40		oprex 3968	. . . . . . . . 9 |- (y + 1) e. V
41	40	isseti 1806	. . . . . . . 8 |- E.x x = (y + 1)
42		ax-17 968	. . . . . . . . . 10 |- (ch -> A.xch)
43	40	hbsbc1v 1940	. . . . . . . . . 10 |- ([(y + 1) / x]ph -> A.x[(y + 1) / x]ph)
44	42, 43	hbbi 1007	. . . . . . . . 9 |- ((ch <-> [(y + 1) / x]ph) -> A.x(ch <-> [(y + 1) / x]ph))
45		nn1suc.3	. . . . . . . . . 10 |- (x = (y + 1) -> (ph <-> ch))
46		sbceq1a 1934	. . . . . . . . . 10 |- (x = (y + 1) -> (ph <-> [(y + 1) / x]ph))
47	45, 46	bitr3d 528	. . . . . . . . 9 |- (x = (y + 1) -> (ch <-> [(y + 1) / x]ph))
48	44, 47	19.23ai 1060	. . . . . . . 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 198	. . . . . 6 |- (y e. NN -> [(y + 1) / x]ph)
51	50	a1d 12	. . . . 5 |- (y e. NN -> (A e. NN -> [(y + 1) / x]ph))
52	21	sbc19.21g 1977	. . . . . 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 200	. . . 4 |- (y e. NN -> [(y + 1) / x](A e. NN -> ph))
55	54	a1d 12	. . 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 5885	. 2 |- (A e. NN -> (A e. NN -> th))
57	56	pm2.43i 64	1 |- (A e. NN -> th)

Syntax hints:   -> wi 3   <-> wb 146   = wceq 953   e. wcel 955  E.wex 977  [wsbc 1166  Vcvv 1802  (class class class)co 3948  1c1 5207   + caddc 5209  NNcn 5268

This theorem is referenced by:  nnleltp1t 5901  ruclem29 7481

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-inf2 4597

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1st 4063  df-2nd 4064  df-1o 4117  df-oadd 4119  df-omul 4120  df-er 4245  df-ec 4247  df-qs 4250  df-ni 4972  df-pli 4973  df-mi 4974  df-lti 4975  df-plpq 5007  df-mpq 5008  df-enq 5009  df-nq 5010  df-plq 5011  df-mq 5012  df-rq 5013  df-ltq 5014  df-1q 5015  df-np 5058  df-1p 5059  df-plp 5060  df-mp 5061  df-ltp 5062  df-plpr 5136  df-mpr 5137  df-enr 5138  df-nr 5139  df-plr 5140  df-mr 5141  df-ltr 5142  df-0r 5143  df-1r 5144  df-m1r 5145  df-c 5212  df-0 5213  df-1 5214  df-i 5215  df-r 5216  df-plus 5217  df-mul 5218  df-sub 5328  df-neg 5330  df-n 5873

Copyright terms: Public domain