Theorem nn1suc 6084

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 1988	. . 3 |- (z = 1 -> ([z / x](A e. NN -> ph) <-> [1 / x](A e. NN -> ph)))
2		sbequ 1266	. . 3 |- (z = y -> ([z / x](A e. NN -> ph) <-> [y / x](A e. NN -> ph)))
3		dfsbcq 1988	. . 3 |- (z = (y + 1) -> ([z / x](A e. NN -> ph) <-> [(y + 1) / x](A e. NN -> ph)))
4		dfsbcq 1988	. . . . . . 7 |- (z = A -> ([z / x]ph <-> [A / x]ph))
5		elex 1865	. . . . . . . . . 10 |- (A e. NN -> E.x x = A)
6		ax-17 1007	. . . . . . . . . . . . 13 |- (z e. A -> A.x z e. A)
7	6	hbsbc1 1994	. . . . . . . . . . . 12 |- ((A e. NN -> [A / x]ph) -> A.x(A e. NN -> [A / x]ph))
8		ax-17 1007	. . . . . . . . . . . 12 |- ((A e. NN -> th) -> A.x(A e. NN -> th))
9	7, 8	hbbi 1046	. . . . . . . . . . 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 1989	. . . . . . . . . . . . 13 |- (x = A -> (ph <-> [A / x]ph))
11		nn1suc.4	. . . . . . . . . . . . 13 |- (x = A -> (ph <-> th))
12	10, 11	bitr3d 533	. . . . . . . . . . . 12 |- (x = A -> ([A / x]ph <-> th))
13	12	imbi2d 615	. . . . . . . . . . 11 |- (x = A -> ((A e. NN -> [A / x]ph) <-> (A e. NN -> th)))
14	9, 13	19.23ai 1100	. . . . . . . . . 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 591	. . . . . . . 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 544	. . . . . 6 |- ((A e. NN /\ z = A) -> ([z / x]ph <-> th))
19	18	expcom 372	. . . . 5 |- (z = A -> (A e. NN -> ([z / x]ph <-> th)))
20	19	pm5.74d 588	. . . 4 |- (z = A -> ((A e. NN -> [z / x]ph) <-> (A e. NN -> th)))
21		ax-17 1007	. . . . 5 |- (A e. NN -> A.x A e. NN)
22	21	sb19.21 1273	. . . 4 |- ([z / x](A e. NN -> ph) <-> (A e. NN -> [z / x]ph))
23	20, 22	syl5bb 535	. . 3 |- (z = A -> ([z / x](A e. NN -> ph) <-> (A e. NN -> th)))
24		1nn 6079	. . . . . . . 8 |- 1 e. NN
25	24	elisseti 1864	. . . . . . 7 |- 1 e. V
26	25	isseti 1861	. . . . . 6 |- E.x x = 1
27	25	hbsbc1v 1995	. . . . . . 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 192	. . . . . . . 8 |- (x = 1 -> ph)
31		sbceq1a 1989	. . . . . . . 8 |- (x = 1 -> (ph <-> [1 / x]ph))
32	30, 31	mpbid 193	. . . . . . 7 |- (x = 1 -> [1 / x]ph)
33	27, 32	19.23ai 1100	. . . . . 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 2035	. . . . 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 188	. . 3 |- [1 / x](A e. NN -> ph)
39		nn1suc.6	. . . . . . 7 |- (y e. NN -> ch)
40		oprex 4041	. . . . . . . . 9 |- (y + 1) e. V
41	40	isseti 1861	. . . . . . . 8 |- E.x x = (y + 1)
42		ax-17 1007	. . . . . . . . . 10 |- (ch -> A.xch)
43	40	hbsbc1v 1995	. . . . . . . . . 10 |- ([(y + 1) / x]ph -> A.x[(y + 1) / x]ph)
44	42, 43	hbbi 1046	. . . . . . . . 9 |- ((ch <-> [(y + 1) / x]ph) -> A.x(ch <-> [(y + 1) / x]ph))
45		nn1suc.3	. . . . . . . . . 10 |- (x = (y + 1) -> (ph <-> ch))
46		sbceq1a 1989	. . . . . . . . . 10 |- (x = (y + 1) -> (ph <-> [(y + 1) / x]ph))
47	45, 46	bitr3d 533	. . . . . . . . 9 |- (x = (y + 1) -> (ch <-> [(y + 1) / x]ph))
48	44, 47	19.23ai 1100	. . . . . . . 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 196	. . . . . 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 2035	. . . . . 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 198	. . . 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 6082	. 2 |- (A e. NN -> (A e. NN -> th))
57	56	pm2.43i 64	1 |- (A e. NN -> th)

Syntax hints:   -> wi 3   <-> wb 144   = wceq 992   e. wcel 994  E.wex 1016  [wsbc 1207  Vcvv 1857  (class class class)co 4021  1c1 5389   + caddc 5391  NNcn 5450

This theorem is referenced by:  nnleltp1 6100  ruclem29 7750

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-9 1001  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-rep 2767  ax-sep 2777  ax-nul 2784  ax-pow 2818  ax-pr 2855  ax-un 3089  ax-inf2 4770

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3or 782  df-3an 783  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-ral 1695  df-rex 1696  df-reu 1697  df-rab 1698  df-v 1858  df-sbc 1987  df-csb 2052  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-pss 2107  df-nul 2333  df-if 2416  df-pw 2459  df-sn 2470  df-pr 2471  df-tp 2473  df-op 2474  df-uni 2570  df-int 2601  df-iun 2635  df-br 2693  df-opab 2741  df-tr 2755  df-eprel 2910  df-id 2913  df-po 2918  df-so 2929  df-fr 2947  df-we 2962  df-ord 2978  df-on 2979  df-lim 2980  df-suc 2981  df-om 3219  df-xp 3265  df-rel 3266  df-cnv 3267  df-co 3268  df-dm 3269  df-rn 3270  df-res 3271  df-ima 3272  df-fun 3273  df-fn 3274  df-f 3275  df-fv 3279  df-opr 4023  df-oprab 4024  df-1st 4140  df-2nd 4141  df-rdg 4233  df-1o 4269  df-oadd 4271  df-omul 4272  df-er 4401  df-ec 4403  df-qs 4406  df-ni 5154  df-pli 5155  df-mi 5156  df-lti 5157  df-plpq 5189  df-mpq 5190  df-enq 5191  df-nq 5192  df-plq 5193  df-mq 5194  df-rq 5195  df-ltq 5196  df-1q 5197  df-np 5240  df-1p 5241  df-plp 5242  df-mp 5243  df-ltp 5244  df-plpr 5318  df-mpr 5319  df-enr 5320  df-nr 5321  df-plr 5322  df-mr 5323  df-ltr 5324  df-0r 5325  df-1r 5326  df-m1r 5327  df-c 5394  df-0 5395  df-1 5396  df-i 5397  df-r 5398  df-plus 5399  df-mul 5400  df-sub 5510  df-neg 5512  df-n 6070

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