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Theorem csbie2 3139
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 1537 . 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 3138 . 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 358   A.wal 1530    = wceq 1632    e. wcel 1696   _Vcvv 2801   [_csb 3094
This theorem is referenced by:  fsumcnv  12252  dfrhm2  15514  mamufval  27546
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-sbc 3005  df-csb 3095
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