ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ectocl Unicode version

Theorem ectocl 6349
Description: Implicit substitution of class for equivalence class. (Contributed by NM, 23-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
Hypotheses
Ref Expression
ectocl.1  |-  S  =  ( B /. R
)
ectocl.2  |-  ( [ x ] R  =  A  ->  ( ph  <->  ps ) )
ectocl.3  |-  ( x  e.  B  ->  ph )
Assertion
Ref Expression
ectocl  |-  ( A  e.  S  ->  ps )
Distinct variable groups:    x, A    x, B    x, R    ps, x
Allowed substitution hints:    ph( x)    S( x)

Proof of Theorem ectocl
StepHypRef Expression
1 tru 1293 . 2  |- T.
2 ectocl.1 . . 3  |-  S  =  ( B /. R
)
3 ectocl.2 . . 3  |-  ( [ x ] R  =  A  ->  ( ph  <->  ps ) )
4 ectocl.3 . . . 4  |-  ( x  e.  B  ->  ph )
54adantl 271 . . 3  |-  ( ( T.  /\  x  e.  B )  ->  ph )
62, 3, 5ectocld 6348 . 2  |-  ( ( T.  /\  A  e.  S )  ->  ps )
71, 6mpan 415 1  |-  ( A  e.  S  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1289   T. wtru 1290    e. wcel 1438   [cec 6280   /.cqs 6281
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-qs 6288
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator