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Theorem wfis2g 25519
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 1630 . 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 25517 1  |-  ( ( R  We  A  /\  R Se  A )  ->  A. y  e.  A  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    e. wcel 1727   A.wral 2711   Se wse 4568    We wwe 4569   Predcpred 25469
This theorem is referenced by:  wfis2  25520  wfr3g  25568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pr 4432
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-br 4238  df-opab 4292  df-po 4532  df-so 4533  df-fr 4570  df-se 4571  df-we 4572  df-xp 4913  df-cnv 4915  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-pred 25470
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