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Theorem sbc2iedv 3059
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 10 . 2  |-  ( ph  ->  A  e.  _V )
3 sbc2iedv.2 . . . 4  |-  B  e. 
_V
43a1i 10 . . 3  |-  ( (
ph  /\  x  =  A )  ->  B  e.  _V )
5 sbc2iedv.3 . . . 4  |-  ( ph  ->  ( ( x  =  A  /\  y  =  B )  ->  ( ps 
<->  ch ) ) )
65impl 603 . . 3  |-  ( ( ( ph  /\  x  =  A )  /\  y  =  B )  ->  ( ps 
<->  ch ) )
74, 6sbcied 3027 . 2  |-  ( (
ph  /\  x  =  A )  ->  ( [. B  /  y ]. ps  <->  ch ) )
82, 7sbcied 3027 1  |-  ( ph  ->  ( [. A  /  x ]. [. B  / 
y ]. ps  <->  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   [.wsbc 2991
This theorem is referenced by:  dfoprab3  6176  sdclem1  26453
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
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