Theorem nnindALT 5900

Description: Principle of Mathematical Induction (inference schema). The last four hypotheses give us the substitution instances we need; the first two are the induction hypothesis and the basis. (This ALT version of nnind 5899 is easier to use with the Proof Assistant since 'assign last' will be applied to the substitution instances first. We may switch to it as the official version.)

Hypotheses

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

Assertion

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

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

Proof of Theorem nnindALT

Step	Hyp	Ref	Expression
1		nnindALT.1	. 2 |- (x = 1 -> (ph <-> ps))
2		nnindALT.2	. 2 |- (x = y -> (ph <-> ch))
3		nnindALT.3	. 2 |- (x = (y + 1) -> (ph <-> th))
4		nnindALT.4	. 2 |- (x = A -> (ph <-> ta))
5		nnindALT.5	. 2 |- ps
6		nnindALT.6	. 2 |- (y e. NN -> (ch -> th))
7	1, 2, 3, 4, 5, 6	nnind 5899	1 |- (A e. NN -> ta)

Syntax hints:   -> wi 3   <-> wb 146   = wceq 955   e. wcel 957  (class class class)co 3960  1c1 5222   + caddc 5224  NNcn 5283

This theorem is referenced by:  fsumcnlem 7972  bcthlem17 7998  bcthlem18 7999

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2690  ax-sep 2700  ax-nul 2707  ax-pow 2739  ax-pr 2776  ax-un 2863  ax-inf2 4612

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-ral 1648  df-rex 1649  df-reu 1650  df-rab 1651  df-v 1810  df-sbc 1940  df-csb 2000  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-pss 2053  df-nul 2279  df-if 2360  df-pw 2400  df-sn 2410  df-pr 2411  df-tp 2413  df-op 2414  df-uni 2501  df-int 2531  df-iun 2565  df-br 2617  df-opab 2664  df-tr 2678  df-eprel 2829  df-id 2832  df-po 2837  df-so 2847  df-fr 2914  df-we 2931  df-ord 2948  df-on 2949  df-lim 2950  df-suc 2951  df-om 3129  df-xp 3181  df-rel 3182  df-cnv 3183  df-co 3184  df-dm 3185  df-rn 3186  df-res 3187  df-ima 3188  df-fun 3189  df-fn 3190  df-f 3191  df-fv 3195  df-rdg 3929  df-opr 3962  df-oprab 3963  df-1st 4076  df-2nd 4077  df-1o 4130  df-oadd 4132  df-omul 4133  df-er 4258  df-ec 4260  df-qs 4263  df-ni 4987  df-pli 4988  df-mi 4989  df-lti 4990  df-plpq 5022  df-mpq 5023  df-enq 5024  df-nq 5025  df-plq 5026  df-mq 5027  df-rq 5028  df-ltq 5029  df-1q 5030  df-np 5073  df-1p 5074  df-plp 5075  df-mp 5076  df-ltp 5077  df-plpr 5151  df-mpr 5152  df-enr 5153  df-nr 5154  df-plr 5155  df-mr 5156  df-ltr 5157  df-0r 5158  df-1r 5159  df-m1r 5160  df-c 5227  df-0 5228  df-1 5229  df-i 5230  df-r 5231  df-plus 5232  df-mul 5233  df-sub 5343  df-neg 5345  df-n 5887

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