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Theorem sbcied2 3158
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 20 . . . 4  |-  ( x  =  A  ->  x  =  A )
3 sbcied2.2 . . . 4  |-  ( ph  ->  A  =  B )
42, 3sylan9eqr 2458 . . 3  |-  ( (
ph  /\  x  =  A )  ->  x  =  B )
5 sbcied2.3 . . 3  |-  ( (
ph  /\  x  =  B )  ->  ( ps 
<->  ch ) )
64, 5syldan 457 . 2  |-  ( (
ph  /\  x  =  A )  ->  ( ps 
<->  ch ) )
71, 6sbcied 3157 1  |-  ( ph  ->  ( [. A  /  x ]. ps  <->  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   [.wsbc 3121
This theorem is referenced by:  iscat  13852  sectffval  13931  issubc  13990  isfunc  14016  ismnd  14647  isnsg  14924  isrng  15623  islbs  16103  isdomn  16309  isassa  16330  opsrval  16490
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-v 2918  df-sbc 3122
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