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Theorem sbcied 2875
Description: Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.)
Hypotheses
Ref Expression
sbcied.1  |-  ( ph  ->  A  e.  V )
sbcied.2  |-  ( (
ph  /\  x  =  A )  ->  ( ps 
<->  ch ) )
Assertion
Ref Expression
sbcied  |-  ( ph  ->  ( [. A  /  x ]. ps  <->  ch )
)
Distinct variable groups:    x, A    ph, x    ch, x
Allowed substitution hints:    ps( x)    V( x)

Proof of Theorem sbcied
StepHypRef Expression
1 sbcied.1 . 2  |-  ( ph  ->  A  e.  V )
2 sbcied.2 . 2  |-  ( (
ph  /\  x  =  A )  ->  ( ps 
<->  ch ) )
3 nfv 1466 . 2  |-  F/ x ph
4 nfvd 1467 . 2  |-  ( ph  ->  F/ x ch )
51, 2, 3, 4sbciedf 2874 1  |-  ( ph  ->  ( [. A  /  x ]. ps  <->  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289    e. wcel 1438   [.wsbc 2840
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-sbc 2841
This theorem is referenced by:  sbcied2  2876  sbc2iedv  2911  sbc3ie  2912  sbcralt  2915  sbcrext  2916  euotd  4081  riota5f  5632
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