MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  csbie2g Structured version   Unicode version

Theorem csbie2g 3289
Description: Conversion of implicit substitution to explicit class substitution. This version of sbcie 3187 avoids a disjointness condition on  x ,  A by substituting twice. (Contributed by Mario Carneiro, 11-Nov-2016.)
Hypotheses
Ref Expression
csbie2g.1  |-  ( x  =  y  ->  B  =  C )
csbie2g.2  |-  ( y  =  A  ->  C  =  D )
Assertion
Ref Expression
csbie2g  |-  ( A  e.  V  ->  [_ A  /  x ]_ B  =  D )
Distinct variable groups:    x, y    y, A    y, B    x, C    y, D
Allowed substitution hints:    A( x)    B( x)    C( y)    D( x)    V( x, y)

Proof of Theorem csbie2g
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 df-csb 3244 . 2  |-  [_ A  /  x ]_ B  =  { z  |  [. A  /  x ]. z  e.  B }
2 csbie2g.1 . . . . 5  |-  ( x  =  y  ->  B  =  C )
32eleq2d 2502 . . . 4  |-  ( x  =  y  ->  (
z  e.  B  <->  z  e.  C ) )
4 csbie2g.2 . . . . 5  |-  ( y  =  A  ->  C  =  D )
54eleq2d 2502 . . . 4  |-  ( y  =  A  ->  (
z  e.  C  <->  z  e.  D ) )
63, 5sbcie2g 3186 . . 3  |-  ( A  e.  V  ->  ( [. A  /  x ]. z  e.  B  <->  z  e.  D ) )
76abbi1dv 2551 . 2  |-  ( A  e.  V  ->  { z  |  [. A  /  x ]. z  e.  B }  =  D )
81, 7syl5eq 2479 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ B  =  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   {cab 2421   [.wsbc 3153   [_csb 3243
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-sbc 3154  df-csb 3244
  Copyright terms: Public domain W3C validator