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Theorem ectocl 6975
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 1331 . 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 454 . . 3  |-  ( (  T.  /\  x  e.  B )  ->  ph )
62, 3, 5ectocld 6974 . 2  |-  ( (  T.  /\  A  e.  S )  ->  ps )
71, 6mpan 653 1  |-  ( A  e.  S  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    T. wtru 1326    = wceq 1653    e. wcel 1726   [cec 6906   /.cqs 6907
This theorem is referenced by:  vitalilem2  19506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-v 2960  df-qs 6914
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