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Theorem csbie2t 3125
Description: Conversion of implicit substitution to explicit substitution into a class (closed form of csbie2 3126). (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 1756 . 2  |-  F/ x A. x A. y ( ( x  =  A  /\  y  =  B )  ->  C  =  D )
2 nfcvd 2420 . 2  |-  ( A. x A. y ( ( x  =  A  /\  y  =  B )  ->  C  =  D )  ->  F/_ x D )
3 csbie2t.1 . . 3  |-  A  e. 
_V
43a1i 10 . 2  |-  ( A. x A. y ( ( x  =  A  /\  y  =  B )  ->  C  =  D )  ->  A  e.  _V )
5 nfa2 1777 . . . 4  |-  F/ y A. x A. y
( ( x  =  A  /\  y  =  B )  ->  C  =  D )
6 nfv 1605 . . . 4  |-  F/ y  x  =  A
75, 6nfan 1771 . . 3  |-  F/ y ( A. x A. y ( ( x  =  A  /\  y  =  B )  ->  C  =  D )  /\  x  =  A )
8 nfcvd 2420 . . 3  |-  ( ( A. x A. y
( ( x  =  A  /\  y  =  B )  ->  C  =  D )  /\  x  =  A )  ->  F/_ y D )
9 csbie2t.2 . . . 4  |-  B  e. 
_V
109a1i 10 . . 3  |-  ( ( A. x A. y
( ( x  =  A  /\  y  =  B )  ->  C  =  D )  /\  x  =  A )  ->  B  e.  _V )
11 sp 1716 . . . . 5  |-  ( A. y ( ( x  =  A  /\  y  =  B )  ->  C  =  D )  ->  (
( x  =  A  /\  y  =  B )  ->  C  =  D ) )
1211sps 1739 . . . 4  |-  ( A. x A. y ( ( x  =  A  /\  y  =  B )  ->  C  =  D )  ->  ( ( x  =  A  /\  y  =  B )  ->  C  =  D ) )
1312impl 603 . . 3  |-  ( ( ( A. x A. y ( ( x  =  A  /\  y  =  B )  ->  C  =  D )  /\  x  =  A )  /\  y  =  B )  ->  C  =  D )
147, 8, 10, 13csbiedf 3118 . 2  |-  ( ( A. x A. y
( ( x  =  A  /\  y  =  B )  ->  C  =  D )  /\  x  =  A )  ->  [_ B  /  y ]_ C  =  D )
151, 2, 4, 14csbiedf 3118 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 358   A.wal 1527    = wceq 1623    e. wcel 1684   _Vcvv 2788   [_csb 3081
This theorem is referenced by:  csbie2  3126
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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