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Theorem ectocld 6461
Description: Implicit substitution of class for equivalence class. (Contributed by Mario Carneiro, 9-Jul-2014.)
Hypotheses
Ref Expression
ectocl.1  |-  S  =  ( B /. R
)
ectocl.2  |-  ( [ x ] R  =  A  ->  ( ph  <->  ps ) )
ectocld.3  |-  ( ( ch  /\  x  e.  B )  ->  ph )
Assertion
Ref Expression
ectocld  |-  ( ( ch  /\  A  e.  S )  ->  ps )
Distinct variable groups:    x, A    x, B    x, R    ps, x    ch, x
Allowed substitution hints:    ph( x)    S( x)

Proof of Theorem ectocld
StepHypRef Expression
1 elqsi 6447 . . . 4  |-  ( A  e.  ( B /. R )  ->  E. x  e.  B  A  =  [ x ] R
)
2 ectocl.1 . . . 4  |-  S  =  ( B /. R
)
31, 2eleq2s 2210 . . 3  |-  ( A  e.  S  ->  E. x  e.  B  A  =  [ x ] R
)
4 ectocld.3 . . . . 5  |-  ( ( ch  /\  x  e.  B )  ->  ph )
5 ectocl.2 . . . . . 6  |-  ( [ x ] R  =  A  ->  ( ph  <->  ps ) )
65eqcoms 2118 . . . . 5  |-  ( A  =  [ x ] R  ->  ( ph  <->  ps )
)
74, 6syl5ibcom 154 . . . 4  |-  ( ( ch  /\  x  e.  B )  ->  ( A  =  [ x ] R  ->  ps )
)
87rexlimdva 2524 . . 3  |-  ( ch 
->  ( E. x  e.  B  A  =  [
x ] R  ->  ps ) )
93, 8syl5 32 . 2  |-  ( ch 
->  ( A  e.  S  ->  ps ) )
109imp 123 1  |-  ( ( ch  /\  A  e.  S )  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1314    e. wcel 1463   E.wrex 2392   [cec 6393   /.cqs 6394
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-qs 6401
This theorem is referenced by:  ectocl  6462  elqsn0m  6463  qsel  6472
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