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Theorem sbcied2 2916
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 2170 . . 3  |-  ( (
ph  /\  x  =  A )  ->  x  =  B )
5 sbcied2.3 . . 3  |-  ( (
ph  /\  x  =  B )  ->  ( ps 
<->  ch ) )
64, 5syldan 278 . 2  |-  ( (
ph  /\  x  =  A )  ->  ( ps 
<->  ch ) )
71, 6sbcied 2915 1  |-  ( ph  ->  ( [. A  /  x ]. ps  <->  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1314    e. wcel 1463   [.wsbc 2880
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-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-sbc 2881
This theorem is referenced by: (None)
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