MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  csbie2 Structured version   Unicode version

Theorem csbie2 3297
Description: Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 27-Aug-2007.)
Hypotheses
Ref Expression
csbie2t.1  |-  A  e. 
_V
csbie2t.2  |-  B  e. 
_V
csbie2.3  |-  ( ( x  =  A  /\  y  =  B )  ->  C  =  D )
Assertion
Ref Expression
csbie2  |-  [_ A  /  x ]_ [_ B  /  y ]_ C  =  D
Distinct variable groups:    x, y, A    x, B, y    x, D, y
Allowed substitution hints:    C( x, y)

Proof of Theorem csbie2
StepHypRef Expression
1 csbie2.3 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  C  =  D )
21gen2 1557 . 2  |-  A. x A. y ( ( x  =  A  /\  y  =  B )  ->  C  =  D )
3 csbie2t.1 . . 3  |-  A  e. 
_V
4 csbie2t.2 . . 3  |-  B  e. 
_V
53, 4csbie2t 3296 . 2  |-  ( A. x A. y ( ( x  =  A  /\  y  =  B )  ->  C  =  D )  ->  [_ A  /  x ]_ [_ B  /  y ]_ C  =  D
)
62, 5ax-mp 8 1  |-  [_ A  /  x ]_ [_ B  /  y ]_ C  =  D
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360   A.wal 1550    = wceq 1653    e. wcel 1726   _Vcvv 2957   [_csb 3252
This theorem is referenced by:  fsumcnv  12558  dfrhm2  15822  fprodcnv  25308  mamufval  27421
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-v 2959  df-sbc 3163  df-csb 3253
  Copyright terms: Public domain W3C validator