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Theorem sbc3ie 3060
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 3027 . 2  |-  ( ( x  =  A  /\  y  =  B )  ->  ( [. C  / 
z ]. ph  <->  ps )
)
81, 2, 7sbc2ie 3058 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 1623    e. wcel 1684   _Vcvv 2788   [.wsbc 2991
This theorem is referenced by:  isdlat  14296  islmod  15631  isibg2  26110  isibcg  26191  rmydioph  27107  hdmap1fval  31987  hdmapfval  32020  hgmapfval  32079
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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