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Theorem sbcied 3027
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 1605 . 2  |-  F/ x ph
4 nfvd 1606 . 2  |-  ( ph  ->  F/ x ch )
51, 2, 3, 4sbciedf 3026 1  |-  ( ph  ->  ( [. A  /  x ]. ps  <->  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   [.wsbc 2991
This theorem is referenced by:  sbcied2  3028  sbc2iedv  3059  sbc3ie  3060  sbcralt  3063  euotd  4267  riota5f  6329  fpwwe2lem12  8263  fpwwe2lem13  8264  issubc  13712  gsumvalx  14451  dmdprd  15236  dprdval  15238  issrng  15615  islmhm  15784  isassa  16056  isphl  16532  istmd  17757  istgp  17760  isnlm  18186  isclm  18562  iscph  18606  iscms  18767  limcfval  19222  abfmpeld  23218  abfmpel  23219
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-sbc 2992
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