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Theorem bnj157 29292
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 29291 . 2  |-  ( ( R  Fr  A  /\  A. x  e.  A  ( ps  ->  ph ) )  ->  A. x  e.  A  ph )
51, 4mpan 653 1  |-  ( A. x  e.  A  ( ps  ->  ph )  ->  A. x  e.  A  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    e. wcel 1726   A.wral 2707   _Vcvv 2958   [.wsbc 3163   class class class wbr 4214    Fr wfr 4540
This theorem is referenced by:  bnj852  29354
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215  df-fr 4543
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