Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  setindis Unicode version

Theorem setindis 9427
Description: Axiom of set induction using implicit substitutions. (Contributed by BJ, 22-Nov-2019.)
Hypotheses
Ref Expression
setindis.nf0  F/
setindis.nf1  F/
setindis.nf2  F/
setindis.nf3  F/
setindis.1
setindis.2
Assertion
Ref Expression
setindis
Distinct variable groups:   ,,   ,
Allowed substitution hints:   (,)   (,,)   (,,)

Proof of Theorem setindis
StepHypRef Expression
1 nfcv 2175 . . . . 5  F/_
2 setindis.nf0 . . . . 5  F/
31, 2nfralxy 2354 . . . 4  F/
4 setindis.nf1 . . . 4  F/
53, 4nfim 1461 . . 3  F/
6 nfcv 2175 . . . . 5  F/_
7 setindis.nf3 . . . . 5  F/
86, 7nfralxy 2354 . . . 4  F/
9 setindis.nf2 . . . 4  F/
108, 9nfim 1461 . . 3  F/
11 raleq 2499 . . . . 5
1211biimprd 147 . . . 4
13 setindis.2 . . . . 5
1413equcoms 1591 . . . 4
1512, 14imim12d 68 . . 3
165, 10, 15cbv3 1627 . 2
17 setindis.1 . . . . . 6
182, 17bj-sbime 9248 . . . . 5
1918ralimi 2378 . . . 4
2019imim1i 54 . . 3
2120alimi 1341 . 2
22 ax-setind 4220 . 2
2316, 21, 223syl 17 1
Colors of variables: wff set class
Syntax hints:   wi 4  wal 1240   F/wnf 1346  wsb 1642  wral 2300
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-setind 4220
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305
This theorem is referenced by:  bj-inf2vnlem4  9433  bj-findis  9439
  Copyright terms: Public domain W3C validator