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Theorem sbc2ie 2829
Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Revised by Mario Carneiro, 19-Dec-2013.)
Hypotheses
Ref Expression
sbc2ie.1  |-  A  e. 
_V
sbc2ie.2  |-  B  e. 
_V
sbc2ie.3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)
Assertion
Ref Expression
sbc2ie  |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  ps )
Distinct variable groups:    x, y, A   
y, B    ps, x, y
Allowed substitution hints:    ph( x, y)    B( x)

Proof of Theorem sbc2ie
StepHypRef Expression
1 sbc2ie.1 . 2  |-  A  e. 
_V
2 sbc2ie.2 . 2  |-  B  e. 
_V
3 nfv 1421 . . 3  |-  F/ x ps
4 nfv 1421 . . 3  |-  F/ y ps
52nfth 1353 . . 3  |-  F/ x  B  e.  _V
6 sbc2ie.3 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)
73, 4, 5, 6sbc2iegf 2828 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( [. A  /  x ]. [. B  / 
y ]. ph  <->  ps )
)
81, 2, 7mp2an 402 1  |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    <-> wb 98    = wceq 1243    e. wcel 1393   _Vcvv 2557   [.wsbc 2764
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sbc 2765
This theorem is referenced by:  sbc3ie  2831
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