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Theorem csbied2 3458
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 2520 . . 3  |-  ( (
ph  /\  x  =  A )  ->  x  =  B )
5 csbied2.3 . . 3  |-  ( (
ph  /\  x  =  B )  ->  C  =  D )
64, 5syldan 470 . 2  |-  ( (
ph  /\  x  =  A )  ->  C  =  D )
71, 6csbied 3457 1  |-  ( ph  ->  [_ A  /  x ]_ C  =  D
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   [_csb 3430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-sbc 3328  df-csb 3431
This theorem is referenced by:  prdsval  14871  cidfval  15092  monfval  15147  idfuval  15291  isnat  15362  fucco  15377  catcval  15501  xpcval  15572  1stfval  15586  2ndfval  15589  prfval  15594  evlf2  15613  curfval  15618  hofval  15647  ipoval  15910  rngcvalOLD  32871  ringcvalOLD  32917
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