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Theorem csbied2 3312
Description: Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
csbied2.1  |-  ( ph  ->  A  e.  V )
csbied2.2  |-  ( ph  ->  A  =  B )
csbied2.3  |-  ( (
ph  /\  x  =  B )  ->  C  =  D )
Assertion
Ref Expression
csbied2  |-  ( ph  ->  [_ A  /  x ]_ C  =  D
)
Distinct variable groups:    x, A    ph, x    x, D
Allowed substitution hints:    B( x)    C( x)    V( x)

Proof of Theorem csbied2
StepHypRef Expression
1 csbied2.1 . 2  |-  ( ph  ->  A  e.  V )
2 id 22 . . . 4  |-  ( x  =  A  ->  x  =  A )
3 csbied2.2 . . . 4  |-  ( ph  ->  A  =  B )
42, 3sylan9eqr 2495 . . 3  |-  ( (
ph  /\  x  =  A )  ->  x  =  B )
5 csbied2.3 . . 3  |-  ( (
ph  /\  x  =  B )  ->  C  =  D )
64, 5syldan 467 . 2  |-  ( (
ph  /\  x  =  A )  ->  C  =  D )
71, 6csbied 3311 1  |-  ( ph  ->  [_ A  /  x ]_ C  =  D
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761   [_csb 3285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-v 2972  df-sbc 3184  df-csb 3286
This theorem is referenced by:  prdsval  14389  cidfval  14610  monfval  14667  idfuval  14782  isnat  14853  fucco  14868  catcval  14960  xpcval  14983  1stfval  14997  2ndfval  15000  prfval  15005  evlf2  15024  curfval  15029  hofval  15058  ipoval  15320
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