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Theorem setinds2f 25160
Description:  _E induction schema, using implicit substitution. (Contributed by Scott Fenton, 10-Mar-2011.) (Revised by Mario Carneiro, 11-Dec-2016.)
Hypotheses
Ref Expression
setinds2f.1  |-  F/ x ps
setinds2f.2  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
setinds2f.3  |-  ( A. y  e.  x  ps  ->  ph )
Assertion
Ref Expression
setinds2f  |-  ph
Distinct variable groups:    x, y    ph, y
Allowed substitution hints:    ph( x)    ps( x, y)

Proof of Theorem setinds2f
StepHypRef Expression
1 sbsbc 3109 . . . . 5  |-  ( [ y  /  x ] ph 
<-> 
[. y  /  x ]. ph )
2 setinds2f.1 . . . . . 6  |-  F/ x ps
3 setinds2f.2 . . . . . 6  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
42, 3sbie 2072 . . . . 5  |-  ( [ y  /  x ] ph 
<->  ps )
51, 4bitr3i 243 . . . 4  |-  ( [. y  /  x ]. ph  <->  ps )
65ralbii 2674 . . 3  |-  ( A. y  e.  x  [. y  /  x ]. ph  <->  A. y  e.  x  ps )
7 setinds2f.3 . . 3  |-  ( A. y  e.  x  ps  ->  ph )
86, 7sylbi 188 . 2  |-  ( A. y  e.  x  [. y  /  x ]. ph  ->  ph )
98setinds 25159 1  |-  ph
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   F/wnf 1550   [wsb 1655   A.wral 2650   [.wsbc 3105
This theorem is referenced by:  setinds2  25161
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-reg 7494  ax-inf2 7530
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-recs 6570  df-rdg 6605
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