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Theorem csbie2t 3424
Description: Conversion of implicit substitution to explicit substitution into a class (closed form of csbie2 3425). (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 1956 . 2  |-  F/ x A. x A. y ( ( x  =  A  /\  y  =  B )  ->  C  =  D )
2 nfcvd 2581 . 2  |-  ( A. x A. y ( ( x  =  A  /\  y  =  B )  ->  C  =  D )  ->  F/_ x D )
3 csbie2t.1 . . 3  |-  A  e. 
_V
43a1i 11 . 2  |-  ( A. x A. y ( ( x  =  A  /\  y  =  B )  ->  C  =  D )  ->  A  e.  _V )
5 nfa2 2013 . . . 4  |-  F/ y A. x A. y
( ( x  =  A  /\  y  =  B )  ->  C  =  D )
6 nfv 1755 . . . 4  |-  F/ y  x  =  A
75, 6nfan 1988 . . 3  |-  F/ y ( A. x A. y ( ( x  =  A  /\  y  =  B )  ->  C  =  D )  /\  x  =  A )
8 nfcvd 2581 . . 3  |-  ( ( A. x A. y
( ( x  =  A  /\  y  =  B )  ->  C  =  D )  /\  x  =  A )  ->  F/_ y D )
9 csbie2t.2 . . . 4  |-  B  e. 
_V
109a1i 11 . . 3  |-  ( ( A. x A. y
( ( x  =  A  /\  y  =  B )  ->  C  =  D )  /\  x  =  A )  ->  B  e.  _V )
11 2sp 1921 . . . 4  |-  ( A. x A. y ( ( x  =  A  /\  y  =  B )  ->  C  =  D )  ->  ( ( x  =  A  /\  y  =  B )  ->  C  =  D ) )
1211impl 624 . . 3  |-  ( ( ( A. x A. y ( ( x  =  A  /\  y  =  B )  ->  C  =  D )  /\  x  =  A )  /\  y  =  B )  ->  C  =  D )
137, 8, 10, 12csbiedf 3416 . 2  |-  ( ( A. x A. y
( ( x  =  A  /\  y  =  B )  ->  C  =  D )  /\  x  =  A )  ->  [_ B  /  y ]_ C  =  D )
141, 2, 4, 13csbiedf 3416 1  |-  ( A. x A. y ( ( x  =  A  /\  y  =  B )  ->  C  =  D )  ->  [_ A  /  x ]_ [_ B  /  y ]_ C  =  D
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370   A.wal 1435    = wceq 1437    e. wcel 1872   _Vcvv 3080   [_csb 3395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-v 3082  df-sbc 3300  df-csb 3396
This theorem is referenced by:  csbie2  3425
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