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Theorem nnindf 26229
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 10439 . . . . . 6  |-  1  e.  NN
2 nnindf.5 . . . . . 6  |-  ps
3 nnindf.1 . . . . . . 7  |-  ( x  =  1  ->  ( ph 
<->  ps ) )
43elrab 3218 . . . . . 6  |-  ( 1  e.  { x  e.  NN  |  ph }  <->  ( 1  e.  NN  /\  ps ) )
51, 2, 4mpbir2an 911 . . . . 5  |-  1  e.  { x  e.  NN  |  ph }
6 elrabi 3215 . . . . . . . 8  |-  ( y  e.  { x  e.  NN  |  ph }  ->  y  e.  NN )
7 peano2nn 10440 . . . . . . . . . . 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 3218 . . . . . . . . 9  |-  ( y  e.  { x  e.  NN  |  ph }  <->  ( y  e.  NN  /\  ch ) )
13 nnindf.3 . . . . . . . . . 10  |-  ( x  =  ( y  +  1 )  ->  ( ph 
<->  th ) )
1413elrab 3218 . . . . . . . . 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 2893 . . . . . 6  |-  A. y  e.  { x  e.  NN  |  ph }  ( y  +  1 )  e. 
{ x  e.  NN  |  ph }
18 nnindf.x . . . . . . . 8  |-  F/ y
ph
19 nfcv 2614 . . . . . . . 8  |-  F/_ y NN
2018, 19nfrab 3002 . . . . . . 7  |-  F/_ y { x  e.  NN  |  ph }
21 nfcv 2614 . . . . . . 7  |-  F/_ w { x  e.  NN  |  ph }
22 nfv 1674 . . . . . . 7  |-  F/ w
( y  +  1 )  e.  { x  e.  NN  |  ph }
23 nfcv 2614 . . . . . . . 8  |-  F/_ y
( w  +  1 )
2423, 20nfel 2626 . . . . . . 7  |-  F/ y ( w  +  1 )  e.  { x  e.  NN  |  ph }
25 oveq1 6202 . . . . . . . 8  |-  ( y  =  w  ->  (
y  +  1 )  =  ( w  + 
1 ) )
2625eleq1d 2521 . . . . . . 7  |-  ( y  =  w  ->  (
( y  +  1 )  e.  { x  e.  NN  |  ph }  <->  ( w  +  1 )  e.  { x  e.  NN  |  ph }
) )
2720, 21, 22, 24, 26cbvralf 3041 . . . . . 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 10431 . . . . 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 3455 . . 3  |-  ( A  e.  NN  ->  A  e.  { x  e.  NN  |  ph } )
32 nnindf.4 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ta ) )
3332elrab 3218 . . 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 1370   F/wnf 1590    e. wcel 1758   A.wral 2796   {crab 2800    C_ wss 3431  (class class class)co 6195   1c1 9389    + caddc 9391   NNcn 10428
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-1cn 9446
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-ov 6198  df-om 6582  df-recs 6937  df-rdg 6971  df-nn 10429
This theorem is referenced by:  nn0min  26230
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