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Theorem sbc3ie 3073
Description: Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Jun-2014.) (Revised by Mario Carneiro, 29-Dec-2014.)
Hypotheses
Ref Expression
sbc3ie.1  |-  A  e. 
_V
sbc3ie.2  |-  B  e. 
_V
sbc3ie.3  |-  C  e. 
_V
sbc3ie.4  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( ph  <->  ps )
)
Assertion
Ref Expression
sbc3ie  |-  ( [. A  /  x ]. [. B  /  y ]. [. C  /  z ]. ph  <->  ps )
Distinct variable groups:    x, y,
z, A    y, B, z    z, C    ps, x, y, z
Allowed substitution hints:    ph( x, y, z)    B( x)    C( x, y)

Proof of Theorem sbc3ie
StepHypRef Expression
1 sbc3ie.1 . 2  |-  A  e. 
_V
2 sbc3ie.2 . 2  |-  B  e. 
_V
3 sbc3ie.3 . . . 4  |-  C  e. 
_V
43a1i 10 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  C  e.  _V )
5 sbc3ie.4 . . . 4  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( ph  <->  ps )
)
653expa 1151 . . 3  |-  ( ( ( x  =  A  /\  y  =  B )  /\  z  =  C )  ->  ( ph 
<->  ps ) )
74, 6sbcied 3040 . 2  |-  ( ( x  =  A  /\  y  =  B )  ->  ( [. C  / 
z ]. ph  <->  ps )
)
81, 2, 7sbc2ie 3071 1  |-  ( [. A  /  x ]. [. B  /  y ]. [. C  /  z ]. ph  <->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   _Vcvv 2801   [.wsbc 3004
This theorem is referenced by:  isdlat  14312  islmod  15647  isibg2  26213  isibcg  26294  rmydioph  27210  hdmap1fval  32609  hdmapfval  32642  hgmapfval  32701
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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