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Theorem tfis2 6673
Description: Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.)
Hypotheses
Ref Expression
tfis2.1  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
tfis2.2  |-  ( x  e.  On  ->  ( A. y  e.  x  ps  ->  ph ) )
Assertion
Ref Expression
tfis2  |-  ( x  e.  On  ->  ph )
Distinct variable groups:    ps, x    ph, y    x, y
Allowed substitution hints:    ph( x)    ps( y)

Proof of Theorem tfis2
StepHypRef Expression
1 nfv 1728 . 2  |-  F/ x ps
2 tfis2.1 . 2  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
3 tfis2.2 . 2  |-  ( x  e.  On  ->  ( A. y  e.  x  ps  ->  ph ) )
41, 2, 3tfis2f 6672 1  |-  ( x  e.  On  ->  ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    e. wcel 1842   A.wral 2753   Oncon0 5409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-tr 4489  df-eprel 4733  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-ord 5412  df-on 5413
This theorem is referenced by:  tfis3  6674  smogt  7070  findcard3  7796  ordiso2  7973  cantnf  8143  cantnfOLD  8165  cfsmolem  8681  fpwwe2lem8  9044  nqereu  9336
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