ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  csbied Unicode version

Theorem csbied 3118
Description: Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 2-Dec-2014.) (Revised by Mario Carneiro, 13-Oct-2016.)
Hypotheses
Ref Expression
csbied.1  |-  ( ph  ->  A  e.  V )
csbied.2  |-  ( (
ph  /\  x  =  A )  ->  B  =  C )
Assertion
Ref Expression
csbied  |-  ( ph  ->  [_ A  /  x ]_ B  =  C
)
Distinct variable groups:    x, A    x, C    ph, x
Allowed substitution hints:    B( x)    V( x)

Proof of Theorem csbied
StepHypRef Expression
1 nfv 1539 . 2  |-  F/ x ph
2 nfcvd 2333 . 2  |-  ( ph  -> 
F/_ x C )
3 csbied.1 . 2  |-  ( ph  ->  A  e.  V )
4 csbied.2 . 2  |-  ( (
ph  /\  x  =  A )  ->  B  =  C )
51, 2, 3, 4csbiedf 3112 1  |-  ( ph  ->  [_ A  /  x ]_ B  =  C
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364    e. wcel 2160   [_csb 3072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-sbc 2978  df-csb 3073
This theorem is referenced by:  csbied2  3119  rspc2vd  3140  fvmptd  5618  seq3f1olemp  10533  fsumgcl  11426  fsum3  11427  fsumshftm  11485  fisum0diag2  11487  fprodseq  11623  fprodeq0  11657  imasival  12783  mulgfvalg  13063  znval  13932  psrval  13944
  Copyright terms: Public domain W3C validator