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Theorem sbc2ie 3071
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 1609 . . 3  |-  F/ x ps
4 nfv 1609 . . 3  |-  F/ y ps
52nfth 1543 . . 3  |-  F/ x  B  e.  _V
6 sbc2ie.3 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)
73, 4, 5, 6sbc2iegf 3070 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( [. A  /  x ]. [. B  / 
y ]. ph  <->  ps )
)
81, 2, 7mp2an 653 1  |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   [.wsbc 3004
This theorem is referenced by:  sbc3ie  3073  isprs  14080  isdrs  14084  istos  14157  bisig0  26165  isibcg  26294  rexrabdioph  26978  rmydioph  27210  rmxdioph  27212  expdiophlem2  27218
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-sbc 3005
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