MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ectocl Unicode version

Theorem ectocl 6727
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 1312 . 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 452 . . 3  |-  ( (  T.  /\  x  e.  B )  ->  ph )
62, 3, 5ectocld 6726 . 2  |-  ( (  T.  /\  A  e.  S )  ->  ps )
71, 6mpan 651 1  |-  ( A  e.  S  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    T. wtru 1307    = wceq 1623    e. wcel 1684   [cec 6658   /.cqs 6659
This theorem is referenced by:  vitalilem2  18964
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-v 2790  df-qs 6666
  Copyright terms: Public domain W3C validator