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Theorem sbc2ie 3344
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 1728 . . 3  |-  F/ x ps
4 nfv 1728 . . 3  |-  F/ y ps
52nfth 1646 . . 3  |-  F/ x  B  e.  _V
6 sbc2ie.3 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)
73, 4, 5, 6sbc2iegf 3343 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( [. A  /  x ]. [. B  / 
y ]. ph  <->  ps )
)
81, 2, 7mp2an 670 1  |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   _Vcvv 3058   [.wsbc 3276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-v 3060  df-sbc 3277
This theorem is referenced by:  sbc3ie  3346  wrd2ind  12757  isprs  15881  isdrs  15885  istos  15987  issrg  17477  isslmd  28183  rexrabdioph  35069  rmydioph  35298  rmxdioph  35300  expdiophlem2  35306
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