MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  csbied2 Unicode version

Theorem csbied2 3254
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 20 . . . 4  |-  ( x  =  A  ->  x  =  A )
3 csbied2.2 . . . 4  |-  ( ph  ->  A  =  B )
42, 3sylan9eqr 2458 . . 3  |-  ( (
ph  /\  x  =  A )  ->  x  =  B )
5 csbied2.3 . . 3  |-  ( (
ph  /\  x  =  B )  ->  C  =  D )
64, 5syldan 457 . 2  |-  ( (
ph  /\  x  =  A )  ->  C  =  D )
71, 6csbied 3253 1  |-  ( ph  ->  [_ A  /  x ]_ C  =  D
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   [_csb 3211
This theorem is referenced by:  prdsval  13633  cidfval  13856  monfval  13913  idfuval  14028  isnat  14099  fucco  14114  catcval  14206  xpcval  14229  1stfval  14243  2ndfval  14246  prfval  14251  evlf2  14270  curfval  14275  hofval  14304  ipoval  14535
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  df-csb 3212
  Copyright terms: Public domain W3C validator