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Theorem csbied2 3137
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 19 . . . 4  |-  ( x  =  A  ->  x  =  A )
3 csbied2.2 . . . 4  |-  ( ph  ->  A  =  B )
42, 3sylan9eqr 2350 . . 3  |-  ( (
ph  /\  x  =  A )  ->  x  =  B )
5 csbied2.3 . . 3  |-  ( (
ph  /\  x  =  B )  ->  C  =  D )
64, 5syldan 456 . 2  |-  ( (
ph  /\  x  =  A )  ->  C  =  D )
71, 6csbied 3136 1  |-  ( ph  ->  [_ A  /  x ]_ C  =  D
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   [_csb 3094
This theorem is referenced by:  prdsval  13371  cidfval  13594  monfval  13651  idfuval  13766  isnat  13837  fucco  13852  catcval  13944  xpcval  13967  1stfval  13981  2ndfval  13984  prfval  13989  evlf2  14008  curfval  14013  hofval  14042
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-sbc 3005  df-csb 3095
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