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

Theorem csbiedf 3417
Description: Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 13-Oct-2016.)
Hypotheses
Ref Expression
csbiedf.1  |-  F/ x ph
csbiedf.2  |-  ( ph  -> 
F/_ x C )
csbiedf.3  |-  ( ph  ->  A  e.  V )
csbiedf.4  |-  ( (
ph  /\  x  =  A )  ->  B  =  C )
Assertion
Ref Expression
csbiedf  |-  ( ph  ->  [_ A  /  x ]_ B  =  C
)
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    B( x)    C( x)    V( x)

Proof of Theorem csbiedf
StepHypRef Expression
1 csbiedf.1 . . 3  |-  F/ x ph
2 csbiedf.4 . . . 4  |-  ( (
ph  /\  x  =  A )  ->  B  =  C )
32ex 434 . . 3  |-  ( ph  ->  ( x  =  A  ->  B  =  C ) )
41, 3alrimi 1816 . 2  |-  ( ph  ->  A. x ( x  =  A  ->  B  =  C ) )
5 csbiedf.3 . . 3  |-  ( ph  ->  A  e.  V )
6 csbiedf.2 . . 3  |-  ( ph  -> 
F/_ x C )
7 csbiebt 3416 . . 3  |-  ( ( A  e.  V  /\  F/_ x C )  -> 
( A. x ( x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C )
)
85, 6, 7syl2anc 661 . 2  |-  ( ph  ->  ( A. x ( x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C )
)
94, 8mpbid 210 1  |-  ( ph  ->  [_ A  /  x ]_ B  =  C
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1368    = wceq 1370   F/wnf 1590    e. wcel 1758   F/_wnfc 2602   [_csb 3396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3080  df-sbc 3295  df-csb 3397
This theorem is referenced by:  csbied  3423  csbie2t  3425  natpropd  15006  fucpropd  15007  gsummptshft  16552  gsummptf1o  26393  sumsnd  29897
  Copyright terms: Public domain W3C validator