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Theorem bnj157 28264
Description: Well-founded induction restricted to a set ( A  e.  _V). The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj157.1  |-  ( ps  <->  A. y  e.  A  ( y R x  ->  [. y  /  x ]. ph ) )
bnj157.2  |-  A  e. 
_V
bnj157.3  |-  R  Fr  A
Assertion
Ref Expression
bnj157  |-  ( A. x  e.  A  ( ps  ->  ph )  ->  A. x  e.  A  ph )
Distinct variable groups:    x, A, y    x, R, y    ph, y
Allowed substitution hints:    ph( x)    ps( x, y)

Proof of Theorem bnj157
StepHypRef Expression
1 bnj157.3 . 2  |-  R  Fr  A
2 bnj157.2 . . 3  |-  A  e. 
_V
3 bnj157.1 . . 3  |-  ( ps  <->  A. y  e.  A  ( y R x  ->  [. y  /  x ]. ph ) )
42, 3bnj110 28263 . 2  |-  ( ( R  Fr  A  /\  A. x  e.  A  ( ps  ->  ph ) )  ->  A. x  e.  A  ph )
51, 4mpan 651 1  |-  ( A. x  e.  A  ( ps  ->  ph )  ->  A. x  e.  A  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    e. wcel 1684   A.wral 2543   _Vcvv 2788   [.wsbc 2991   class class class wbr 4023    Fr wfr 4349
This theorem is referenced by:  bnj852  28326
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-fr 4352
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