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Theorem sbcied2 2877
Description: Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.)
Hypotheses
Ref Expression
sbcied2.1  |-  ( ph  ->  A  e.  V )
sbcied2.2  |-  ( ph  ->  A  =  B )
sbcied2.3  |-  ( (
ph  /\  x  =  B )  ->  ( ps 
<->  ch ) )
Assertion
Ref Expression
sbcied2  |-  ( ph  ->  ( [. A  /  x ]. ps  <->  ch )
)
Distinct variable groups:    x, A    ph, x    ch, x
Allowed substitution hints:    ps( x)    B( x)    V( x)

Proof of Theorem sbcied2
StepHypRef Expression
1 sbcied2.1 . 2  |-  ( ph  ->  A  e.  V )
2 id 19 . . . 4  |-  ( x  =  A  ->  x  =  A )
3 sbcied2.2 . . . 4  |-  ( ph  ->  A  =  B )
42, 3sylan9eqr 2143 . . 3  |-  ( (
ph  /\  x  =  A )  ->  x  =  B )
5 sbcied2.3 . . 3  |-  ( (
ph  /\  x  =  B )  ->  ( ps 
<->  ch ) )
64, 5syldan 277 . 2  |-  ( (
ph  /\  x  =  A )  ->  ( ps 
<->  ch ) )
71, 6sbcied 2876 1  |-  ( ph  ->  ( [. A  /  x ]. ps  <->  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1290    e. wcel 1439   [.wsbc 2841
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-sbc 2842
This theorem is referenced by: (None)
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