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Theorem sbcied2 3325
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 22 . . . 4  |-  ( x  =  A  ->  x  =  A )
3 sbcied2.2 . . . 4  |-  ( ph  ->  A  =  B )
42, 3sylan9eqr 2514 . . 3  |-  ( (
ph  /\  x  =  A )  ->  x  =  B )
5 sbcied2.3 . . 3  |-  ( (
ph  /\  x  =  B )  ->  ( ps 
<->  ch ) )
64, 5syldan 470 . 2  |-  ( (
ph  /\  x  =  A )  ->  ( ps 
<->  ch ) )
71, 6sbcied 3324 1  |-  ( ph  ->  ( [. A  /  x ]. ps  <->  ch )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   [.wsbc 3287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-v 3073  df-sbc 3288
This theorem is referenced by:  iscat  14721  sectffval  14800  issubc  14859  isfunc  14885  ismnd  15528  isnsg  15821  isrng  16764  islbs  17272  isdomn  17481  isassa  17502  opsrval  17672
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