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Theorem nnindf 27278
Description: Principle of Mathematical Induction, using a bound-variable hypothesis instead of distinct variables. (Contributed by Thierry Arnoux, 6-May-2018.)
Hypotheses
Ref Expression
nnindf.x  |-  F/ y
ph
nnindf.1  |-  ( x  =  1  ->  ( ph 
<->  ps ) )
nnindf.2  |-  ( x  =  y  ->  ( ph 
<->  ch ) )
nnindf.3  |-  ( x  =  ( y  +  1 )  ->  ( ph 
<->  th ) )
nnindf.4  |-  ( x  =  A  ->  ( ph 
<->  ta ) )
nnindf.5  |-  ps
nnindf.6  |-  ( y  e.  NN  ->  ( ch  ->  th ) )
Assertion
Ref Expression
nnindf  |-  ( A  e.  NN  ->  ta )
Distinct variable groups:    x, y    x, A    ch, x    ps, x    ta, x    th, x
Allowed substitution hints:    ph( x, y)    ps( y)    ch( y)    th( y)    ta( y)    A( y)

Proof of Theorem nnindf
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 1nn 10543 . . . . . 6  |-  1  e.  NN
2 nnindf.5 . . . . . 6  |-  ps
3 nnindf.1 . . . . . . 7  |-  ( x  =  1  ->  ( ph 
<->  ps ) )
43elrab 3261 . . . . . 6  |-  ( 1  e.  { x  e.  NN  |  ph }  <->  ( 1  e.  NN  /\  ps ) )
51, 2, 4mpbir2an 918 . . . . 5  |-  1  e.  { x  e.  NN  |  ph }
6 elrabi 3258 . . . . . . . 8  |-  ( y  e.  { x  e.  NN  |  ph }  ->  y  e.  NN )
7 peano2nn 10544 . . . . . . . . . . 11  |-  ( y  e.  NN  ->  (
y  +  1 )  e.  NN )
87a1d 25 . . . . . . . . . 10  |-  ( y  e.  NN  ->  (
y  e.  NN  ->  ( y  +  1 )  e.  NN ) )
9 nnindf.6 . . . . . . . . . 10  |-  ( y  e.  NN  ->  ( ch  ->  th ) )
108, 9anim12d 563 . . . . . . . . 9  |-  ( y  e.  NN  ->  (
( y  e.  NN  /\ 
ch )  ->  (
( y  +  1 )  e.  NN  /\  th ) ) )
11 nnindf.2 . . . . . . . . . 10  |-  ( x  =  y  ->  ( ph 
<->  ch ) )
1211elrab 3261 . . . . . . . . 9  |-  ( y  e.  { x  e.  NN  |  ph }  <->  ( y  e.  NN  /\  ch ) )
13 nnindf.3 . . . . . . . . . 10  |-  ( x  =  ( y  +  1 )  ->  ( ph 
<->  th ) )
1413elrab 3261 . . . . . . . . 9  |-  ( ( y  +  1 )  e.  { x  e.  NN  |  ph }  <->  ( ( y  +  1 )  e.  NN  /\  th ) )
1510, 12, 143imtr4g 270 . . . . . . . 8  |-  ( y  e.  NN  ->  (
y  e.  { x  e.  NN  |  ph }  ->  ( y  +  1 )  e.  { x  e.  NN  |  ph }
) )
166, 15mpcom 36 . . . . . . 7  |-  ( y  e.  { x  e.  NN  |  ph }  ->  ( y  +  1 )  e.  { x  e.  NN  |  ph }
)
1716rgen 2824 . . . . . 6  |-  A. y  e.  { x  e.  NN  |  ph }  ( y  +  1 )  e. 
{ x  e.  NN  |  ph }
18 nnindf.x . . . . . . . 8  |-  F/ y
ph
19 nfcv 2629 . . . . . . . 8  |-  F/_ y NN
2018, 19nfrab 3043 . . . . . . 7  |-  F/_ y { x  e.  NN  |  ph }
21 nfcv 2629 . . . . . . 7  |-  F/_ w { x  e.  NN  |  ph }
22 nfv 1683 . . . . . . 7  |-  F/ w
( y  +  1 )  e.  { x  e.  NN  |  ph }
23 nfcv 2629 . . . . . . . 8  |-  F/_ y
( w  +  1 )
2423, 20nfel 2642 . . . . . . 7  |-  F/ y ( w  +  1 )  e.  { x  e.  NN  |  ph }
25 oveq1 6289 . . . . . . . 8  |-  ( y  =  w  ->  (
y  +  1 )  =  ( w  + 
1 ) )
2625eleq1d 2536 . . . . . . 7  |-  ( y  =  w  ->  (
( y  +  1 )  e.  { x  e.  NN  |  ph }  <->  ( w  +  1 )  e.  { x  e.  NN  |  ph }
) )
2720, 21, 22, 24, 26cbvralf 3082 . . . . . 6  |-  ( A. y  e.  { x  e.  NN  |  ph } 
( y  +  1 )  e.  { x  e.  NN  |  ph }  <->  A. w  e.  { x  e.  NN  |  ph } 
( w  +  1 )  e.  { x  e.  NN  |  ph }
)
2817, 27mpbi 208 . . . . 5  |-  A. w  e.  { x  e.  NN  |  ph }  ( w  +  1 )  e. 
{ x  e.  NN  |  ph }
29 peano5nni 10535 . . . . 5  |-  ( ( 1  e.  { x  e.  NN  |  ph }  /\  A. w  e.  {
x  e.  NN  |  ph }  ( w  + 
1 )  e.  {
x  e.  NN  |  ph } )  ->  NN  C_ 
{ x  e.  NN  |  ph } )
305, 28, 29mp2an 672 . . . 4  |-  NN  C_  { x  e.  NN  |  ph }
3130sseli 3500 . . 3  |-  ( A  e.  NN  ->  A  e.  { x  e.  NN  |  ph } )
32 nnindf.4 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ta ) )
3332elrab 3261 . . 3  |-  ( A  e.  { x  e.  NN  |  ph }  <->  ( A  e.  NN  /\  ta ) )
3431, 33sylib 196 . 2  |-  ( A  e.  NN  ->  ( A  e.  NN  /\  ta ) )
3534simprd 463 1  |-  ( A  e.  NN  ->  ta )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379   F/wnf 1599    e. wcel 1767   A.wral 2814   {crab 2818    C_ wss 3476  (class class class)co 6282   1c1 9489    + caddc 9491   NNcn 10532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-1cn 9546
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-om 6679  df-recs 7039  df-rdg 7073  df-nn 10533
This theorem is referenced by:  nn0min  27279
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