Theorem nn1suc 5939

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 1943	. . 3 |- (z = 1 -> ([z / x](A e. NN -> ph) <-> [1 / x](A e. NN -> ph)))
2		sbequ 1229	. . 3 |- (z = y -> ([z / x](A e. NN -> ph) <-> [y / x](A e. NN -> ph)))
3		dfsbcq 1943	. . 3 |- (z = (y + 1) -> ([z / x](A e. NN -> ph) <-> [(y + 1) / x](A e. NN -> ph)))
4		dfsbcq 1943	. . . . . . 7 |- (z = A -> ([z / x]ph <-> [A / x]ph))
5		elex 1819	. . . . . . . . . 10 |- (A e. NN -> E.x x = A)
6		ax-17 971	. . . . . . . . . . . . 13 |- (z e. A -> A.x z e. A)
7	6	hbsbc1 1949	. . . . . . . . . . . 12 |- ((A e. NN -> [A / x]ph) -> A.x(A e. NN -> [A / x]ph))
8		ax-17 971	. . . . . . . . . . . 12 |- ((A e. NN -> th) -> A.x(A e. NN -> th))
9	7, 8	hbbi 1010	. . . . . . . . . . 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 1944	. . . . . . . . . . . . 13 |- (x = A -> (ph <-> [A / x]ph))
11		nn1suc.4	. . . . . . . . . . . . 13 |- (x = A -> (ph <-> th))
12	10, 11	bitr3d 530	. . . . . . . . . . . 12 |- (x = A -> ([A / x]ph <-> th))
13	12	imbi2d 612	. . . . . . . . . . 11 |- (x = A -> ((A e. NN -> [A / x]ph) <-> (A e. NN -> th)))
14	9, 13	19.23ai 1064	. . . . . . . . . 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 588	. . . . . . . 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 541	. . . . . 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 585	. . . 4 |- (z = A -> ((A e. NN -> [z / x]ph) <-> (A e. NN -> th)))
21		ax-17 971	. . . . 5 |- (A e. NN -> A.x A e. NN)
22	21	sb19.21 1236	. . . 4 |- ([z / x](A e. NN -> ph) <-> (A e. NN -> [z / x]ph))
23	20, 22	syl5bb 532	. . 3 |- (z = A -> ([z / x](A e. NN -> ph) <-> (A e. NN -> th)))
24		1nn 5934	. . . . . . . 8 |- 1 e. NN
25	24	elisseti 1818	. . . . . . 7 |- 1 e. V
26	25	isseti 1815	. . . . . 6 |- E.x x = 1
27	25	hbsbc1v 1950	. . . . . . 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 1944	. . . . . . . 8 |- (x = 1 -> (ph <-> [1 / x]ph))
32	30, 31	mpbid 195	. . . . . . 7 |- (x = 1 -> [1 / x]ph)
33	27, 32	19.23ai 1064	. . . . . 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 1987	. . . . 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 3983	. . . . . . . . 9 |- (y + 1) e. V
41	40	isseti 1815	. . . . . . . 8 |- E.x x = (y + 1)
42		ax-17 971	. . . . . . . . . 10 |- (ch -> A.xch)
43	40	hbsbc1v 1950	. . . . . . . . . 10 |- ([(y + 1) / x]ph -> A.x[(y + 1) / x]ph)
44	42, 43	hbbi 1010	. . . . . . . . 9 |- ((ch <-> [(y + 1) / x]ph) -> A.x(ch <-> [(y + 1) / x]ph))
45		nn1suc.3	. . . . . . . . . 10 |- (x = (y + 1) -> (ph <-> ch))
46		sbceq1a 1944	. . . . . . . . . 10 |- (x = (y + 1) -> (ph <-> [(y + 1) / x]ph))
47	45, 46	bitr3d 530	. . . . . . . . 9 |- (x = (y + 1) -> (ch <-> [(y + 1) / x]ph))
48	44, 47	19.23ai 1064	. . . . . . . 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 1987	. . . . . 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 5937	. 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 956   e. wcel 958  E.wex 980  [wsbc 1170  Vcvv 1811  (class class class)co 3963  1c1 5235   + caddc 5237  NNcn 5296

This theorem is referenced by:  nnleltp1t 5954  ruclem29 7538

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-inf2 4625

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fv 3198  df-rdg 3932  df-opr 3965  df-oprab 3966  df-1st 4079  df-2nd 4080  df-1o 4133  df-oadd 4135  df-omul 4136  df-er 4261  df-ec 4263  df-qs 4266  df-ni 5000  df-pli 5001  df-mi 5002  df-lti 5003  df-plpq 5035  df-mpq 5036  df-enq 5037  df-nq 5038  df-plq 5039  df-mq 5040  df-rq 5041  df-ltq 5042  df-1q 5043  df-np 5086  df-1p 5087  df-plp 5088  df-mp 5089  df-ltp 5090  df-plpr 5164  df-mpr 5165  df-enr 5166  df-nr 5167  df-plr 5168  df-mr 5169  df-ltr 5170  df-0r 5171  df-1r 5172  df-m1r 5173  df-c 5240  df-0 5241  df-1 5242  df-i 5243  df-r 5244  df-plus 5245  df-mul 5246  df-sub 5356  df-neg 5358  df-n 5925

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