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Theorem ectocld 6963
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 6950 . . . 4  |-  ( A  e.  ( B /. R )  ->  E. x  e.  B  A  =  [ x ] R
)
2 ectocl.1 . . . 4  |-  S  =  ( B /. R
)
31, 2eleq2s 2527 . . 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 2438 . . . . 5  |-  ( A  =  [ x ] R  ->  ( ph  <->  ps )
)
74, 6syl5ibcom 212 . . . 4  |-  ( ( ch  /\  x  e.  B )  ->  ( A  =  [ x ] R  ->  ps )
)
87rexlimdva 2822 . . 3  |-  ( ch 
->  ( E. x  e.  B  A  =  [
x ] R  ->  ps ) )
93, 8syl5 30 . 2  |-  ( ch 
->  ( A  e.  S  ->  ps ) )
109imp 419 1  |-  ( ( ch  /\  A  e.  S )  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2698   [cec 6895   /.cqs 6896
This theorem is referenced by:  ectocl  6964  elqsn0  6965  qsdisj  6973  qsel  6975  eqgen  14985  orbsta  15082  sylow1lem3  15226  sylow2alem2  15244  sylow2a  15245  sylow2blem2  15247  frgpup1  15399  frgpup3lem  15401  divscrng  16303  pi1xfr  19072  pi1coghm  19078  vitalilem3  19494
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-v 2950  df-qs 6903
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