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Theorem sbc3ie 2900
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 9 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  C  e.  _V )
5 sbc3ie.4 . . . 4  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( ph  <->  ps )
)
653expa 1141 . . 3  |-  ( ( ( x  =  A  /\  y  =  B )  /\  z  =  C )  ->  ( ph 
<->  ps ) )
74, 6sbcied 2863 . 2  |-  ( ( x  =  A  /\  y  =  B )  ->  ( [. C  / 
z ]. ph  <->  ps )
)
81, 2, 7sbc2ie 2898 1  |-  ( [. A  /  x ]. [. B  /  y ]. [. C  /  z ]. ph  <->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 922    = wceq 1287    e. wcel 1436   _Vcvv 2614   [.wsbc 2828
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2616  df-sbc 2829
This theorem is referenced by: (None)
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