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Theorem wfis2g 25427
Description: Well-Founded Induction Schema, using implicit substitution. (Contributed by Scott Fenton, 11-Feb-2011.)
Hypotheses
Ref Expression
wfis2g.1  |-  ( y  =  z  ->  ( ph 
<->  ps ) )
wfis2g.2  |-  ( y  e.  A  ->  ( A. z  e.  Pred  ( R ,  A , 
y ) ps  ->  ph ) )
Assertion
Ref Expression
wfis2g  |-  ( ( R  We  A  /\  R Se  A )  ->  A. y  e.  A  ph )
Distinct variable groups:    y, A, z    ph, z    ps, y    y, R, z
Allowed substitution hints:    ph( y)    ps( z)

Proof of Theorem wfis2g
StepHypRef Expression
1 nfv 1626 . 2  |-  F/ y ps
2 wfis2g.1 . 2  |-  ( y  =  z  ->  ( ph 
<->  ps ) )
3 wfis2g.2 . 2  |-  ( y  e.  A  ->  ( A. z  e.  Pred  ( R ,  A , 
y ) ps  ->  ph ) )
41, 2, 3wfis2fg 25425 1  |-  ( ( R  We  A  /\  R Se  A )  ->  A. y  e.  A  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1721   A.wral 2666   Se wse 4499    We wwe 4500   Predcpred 25381
This theorem is referenced by:  wfis2  25428  wfr3g  25469
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 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-xp 4843  df-cnv 4845  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 25382
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