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Theorem sbc3ie 2912
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 1143 . . 3  |-  ( ( ( x  =  A  /\  y  =  B )  /\  z  =  C )  ->  ( ph 
<->  ps ) )
74, 6sbcied 2875 . 2  |-  ( ( x  =  A  /\  y  =  B )  ->  ( [. C  / 
z ]. ph  <->  ps )
)
81, 2, 7sbc2ie 2910 1  |-  ( [. A  /  x ]. [. B  /  y ]. [. C  /  z ]. ph  <->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 924    = wceq 1289    e. wcel 1438   _Vcvv 2619   [.wsbc 2840
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-sbc 2841
This theorem is referenced by: (None)
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