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Theorem ectocl 6489
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 1335 . 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 275 . . 3  |-  ( ( T.  /\  x  e.  B )  ->  ph )
62, 3, 5ectocld 6488 . 2  |-  ( ( T.  /\  A  e.  S )  ->  ps )
71, 6mpan 420 1  |-  ( A  e.  S  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1331   T. wtru 1332    e. wcel 1480   [cec 6420   /.cqs 6421
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-qs 6428
This theorem is referenced by: (None)
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