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Theorem csbie2t 3105
Description: Conversion of implicit substitution to explicit substitution into a class (closed form of csbie2 3106). (Contributed by NM, 3-Sep-2007.) (Revised by Mario Carneiro, 13-Oct-2016.)
Hypotheses
Ref Expression
csbie2t.1  |-  A  e. 
_V
csbie2t.2  |-  B  e. 
_V
Assertion
Ref Expression
csbie2t  |-  ( A. x A. y ( ( x  =  A  /\  y  =  B )  ->  C  =  D )  ->  [_ 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 csbie2t
StepHypRef Expression
1 nfa1 1541 . 2  |-  F/ x A. x A. y ( ( x  =  A  /\  y  =  B )  ->  C  =  D )
2 nfcvd 2320 . 2  |-  ( A. x A. y ( ( x  =  A  /\  y  =  B )  ->  C  =  D )  ->  F/_ x D )
3 csbie2t.1 . . 3  |-  A  e. 
_V
43a1i 9 . 2  |-  ( A. x A. y ( ( x  =  A  /\  y  =  B )  ->  C  =  D )  ->  A  e.  _V )
5 nfa2 1579 . . . 4  |-  F/ y A. x A. y
( ( x  =  A  /\  y  =  B )  ->  C  =  D )
6 nfv 1528 . . . 4  |-  F/ y  x  =  A
75, 6nfan 1565 . . 3  |-  F/ y ( A. x A. y ( ( x  =  A  /\  y  =  B )  ->  C  =  D )  /\  x  =  A )
8 nfcvd 2320 . . 3  |-  ( ( A. x A. y
( ( x  =  A  /\  y  =  B )  ->  C  =  D )  /\  x  =  A )  ->  F/_ y D )
9 csbie2t.2 . . . 4  |-  B  e. 
_V
109a1i 9 . . 3  |-  ( ( A. x A. y
( ( x  =  A  /\  y  =  B )  ->  C  =  D )  /\  x  =  A )  ->  B  e.  _V )
11 sp 1511 . . . . 5  |-  ( A. y ( ( x  =  A  /\  y  =  B )  ->  C  =  D )  ->  (
( x  =  A  /\  y  =  B )  ->  C  =  D ) )
1211sps 1537 . . . 4  |-  ( A. x A. y ( ( x  =  A  /\  y  =  B )  ->  C  =  D )  ->  ( ( x  =  A  /\  y  =  B )  ->  C  =  D ) )
1312impl 380 . . 3  |-  ( ( ( A. x A. y ( ( x  =  A  /\  y  =  B )  ->  C  =  D )  /\  x  =  A )  /\  y  =  B )  ->  C  =  D )
147, 8, 10, 13csbiedf 3097 . 2  |-  ( ( A. x A. y
( ( x  =  A  /\  y  =  B )  ->  C  =  D )  /\  x  =  A )  ->  [_ B  /  y ]_ C  =  D )
151, 2, 4, 14csbiedf 3097 1  |-  ( A. x A. y ( ( x  =  A  /\  y  =  B )  ->  C  =  D )  ->  [_ A  /  x ]_ [_ B  /  y ]_ C  =  D
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104   A.wal 1351    = wceq 1353    e. wcel 2148   _Vcvv 2737   [_csb 3057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-sbc 2963  df-csb 3058
This theorem is referenced by:  csbie2  3106
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