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Theorem sbc2iedv 2895
Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Proof shortened by Mario Carneiro, 18-Oct-2016.)
Hypotheses
Ref Expression
sbc2iedv.1  |-  A  e. 
_V
sbc2iedv.2  |-  B  e. 
_V
sbc2iedv.3  |-  ( ph  ->  ( ( x  =  A  /\  y  =  B )  ->  ( ps 
<->  ch ) ) )
Assertion
Ref Expression
sbc2iedv  |-  ( ph  ->  ( [. A  /  x ]. [. B  / 
y ]. ps  <->  ch )
)
Distinct variable groups:    x, y, A   
y, B    ph, x, y    ch, x, y
Allowed substitution hints:    ps( x, y)    B( x)

Proof of Theorem sbc2iedv
StepHypRef Expression
1 sbc2iedv.1 . . 3  |-  A  e. 
_V
21a1i 9 . 2  |-  ( ph  ->  A  e.  _V )
3 sbc2iedv.2 . . . 4  |-  B  e. 
_V
43a1i 9 . . 3  |-  ( (
ph  /\  x  =  A )  ->  B  e.  _V )
5 sbc2iedv.3 . . . 4  |-  ( ph  ->  ( ( x  =  A  /\  y  =  B )  ->  ( ps 
<->  ch ) ) )
65impl 372 . . 3  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  ( ps 
<->  ch ) )
74, 6sbcied 2859 . 2  |-  ( (
ph  /\  x  =  A )  ->  ( [. B  /  y ]. ps  <->  ch ) )
82, 7sbcied 2859 1  |-  ( ph  ->  ( [. A  /  x ]. [. B  / 
y ]. ps  <->  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434   _Vcvv 2610   [.wsbc 2824
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-sbc 2825
This theorem is referenced by:  dfoprab3  5869
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