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Theorem ectocl 6814
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 1321 . 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 6813 . 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 1316    = wceq 1642    e. wcel 1710   [cec 6745   /.cqs 6746
This theorem is referenced by:  vitalilem2  19068
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ral 2624  df-rex 2625  df-v 2866  df-qs 6753
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