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Theorem ectocld 6742
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 6729 . . . 4  |-  ( A  e.  ( B /. R )  ->  E. x  e.  B  A  =  [ x ] R
)
2 ectocl.1 . . . 4  |-  S  =  ( B /. R
)
31, 2eleq2s 2388 . . 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 2299 . . . . 5  |-  ( A  =  [ x ] R  ->  ( ph  <->  ps )
)
74, 6syl5ibcom 211 . . . 4  |-  ( ( ch  /\  x  e.  B )  ->  ( A  =  [ x ] R  ->  ps )
)
87rexlimdva 2680 . . 3  |-  ( ch 
->  ( E. x  e.  B  A  =  [
x ] R  ->  ps ) )
93, 8syl5 28 . 2  |-  ( ch 
->  ( A  e.  S  ->  ps ) )
109imp 418 1  |-  ( ( ch  /\  A  e.  S )  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557   [cec 6674   /.cqs 6675
This theorem is referenced by:  ectocl  6743  elqsn0  6744  qsdisj  6752  qsel  6754  eqgen  14686  orbsta  14783  sylow1lem3  14927  sylow2alem2  14945  sylow2a  14946  sylow2blem2  14948  frgpup1  15100  frgpup3lem  15102  divscrng  16008  pi1xfr  18569  pi1coghm  18575  vitalilem3  18981
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-v 2803  df-qs 6682
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