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Theorem nnindf 27844
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 10542 . . . . . 6  |-  1  e.  NN
2 nnindf.5 . . . . . 6  |-  ps
3 nnindf.1 . . . . . . 7  |-  ( x  =  1  ->  ( ph 
<->  ps ) )
43elrab 3254 . . . . . 6  |-  ( 1  e.  { x  e.  NN  |  ph }  <->  ( 1  e.  NN  /\  ps ) )
51, 2, 4mpbir2an 918 . . . . 5  |-  1  e.  { x  e.  NN  |  ph }
6 elrabi 3251 . . . . . . . 8  |-  ( y  e.  { x  e.  NN  |  ph }  ->  y  e.  NN )
7 peano2nn 10543 . . . . . . . . . . 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 561 . . . . . . . . 9  |-  ( y  e.  NN  ->  (
( y  e.  NN  /\ 
ch )  ->  (
( y  +  1 )  e.  NN  /\  th ) ) )
11 nnindf.2 . . . . . . . . . 10  |-  ( x  =  y  ->  ( ph 
<->  ch ) )
1211elrab 3254 . . . . . . . . 9  |-  ( y  e.  { x  e.  NN  |  ph }  <->  ( y  e.  NN  /\  ch ) )
13 nnindf.3 . . . . . . . . . 10  |-  ( x  =  ( y  +  1 )  ->  ( ph 
<->  th ) )
1413elrab 3254 . . . . . . . . 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 2814 . . . . . 6  |-  A. y  e.  { x  e.  NN  |  ph }  ( y  +  1 )  e. 
{ x  e.  NN  |  ph }
18 nnindf.x . . . . . . . 8  |-  F/ y
ph
19 nfcv 2616 . . . . . . . 8  |-  F/_ y NN
2018, 19nfrab 3036 . . . . . . 7  |-  F/_ y { x  e.  NN  |  ph }
21 nfcv 2616 . . . . . . 7  |-  F/_ w { x  e.  NN  |  ph }
22 nfv 1712 . . . . . . 7  |-  F/ w
( y  +  1 )  e.  { x  e.  NN  |  ph }
2320nfel2 2634 . . . . . . 7  |-  F/ y ( w  +  1 )  e.  { x  e.  NN  |  ph }
24 oveq1 6277 . . . . . . . 8  |-  ( y  =  w  ->  (
y  +  1 )  =  ( w  + 
1 ) )
2524eleq1d 2523 . . . . . . 7  |-  ( y  =  w  ->  (
( y  +  1 )  e.  { x  e.  NN  |  ph }  <->  ( w  +  1 )  e.  { x  e.  NN  |  ph }
) )
2620, 21, 22, 23, 25cbvralf 3075 . . . . . 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 }
)
2717, 26mpbi 208 . . . . 5  |-  A. w  e.  { x  e.  NN  |  ph }  ( w  +  1 )  e. 
{ x  e.  NN  |  ph }
28 peano5nni 10534 . . . . 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 } )
295, 27, 28mp2an 670 . . . 4  |-  NN  C_  { x  e.  NN  |  ph }
3029sseli 3485 . . 3  |-  ( A  e.  NN  ->  A  e.  { x  e.  NN  |  ph } )
31 nnindf.4 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ta ) )
3231elrab 3254 . . 3  |-  ( A  e.  { x  e.  NN  |  ph }  <->  ( A  e.  NN  /\  ta ) )
3330, 32sylib 196 . 2  |-  ( A  e.  NN  ->  ( A  e.  NN  /\  ta ) )
3433simprd 461 1  |-  ( A  e.  NN  ->  ta )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398   F/wnf 1621    e. wcel 1823   A.wral 2804   {crab 2808    C_ wss 3461  (class class class)co 6270   1c1 9482    + caddc 9484   NNcn 10531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-1cn 9539
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-om 6674  df-recs 7034  df-rdg 7068  df-nn 10532
This theorem is referenced by:  nn0min  27845
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