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Theorem csbie2 3126
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 1534 . 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 3125 . 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 1527    = wceq 1623    e. wcel 1684   _Vcvv 2788   [_csb 3081
This theorem is referenced by:  fsumcnv  12236  dfrhm2  15498  mamufval  27443
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-sbc 2992  df-csb 3082
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