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Theorem sbc2iegf 2909
Description: Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Dec-2013.)
Hypotheses
Ref Expression
sbc2iegf.1  |-  F/ x ps
sbc2iegf.2  |-  F/ y ps
sbc2iegf.3  |-  F/ x  B  e.  W
sbc2iegf.4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)
Assertion
Ref Expression
sbc2iegf  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( [. A  /  x ]. [. B  / 
y ]. ph  <->  ps )
)
Distinct variable groups:    x, y, A   
y, B    x, V    y, W
Allowed substitution hints:    ph( x, y)    ps( x, y)    B( x)    V( y)    W( x)

Proof of Theorem sbc2iegf
StepHypRef Expression
1 simpl 107 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  A  e.  V )
2 simpl 107 . . . 4  |-  ( ( B  e.  W  /\  x  =  A )  ->  B  e.  W )
3 sbc2iegf.4 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)
43adantll 460 . . . 4  |-  ( ( ( B  e.  W  /\  x  =  A
)  /\  y  =  B )  ->  ( ph 
<->  ps ) )
5 nfv 1466 . . . 4  |-  F/ y ( B  e.  W  /\  x  =  A
)
6 sbc2iegf.2 . . . . 5  |-  F/ y ps
76a1i 9 . . . 4  |-  ( ( B  e.  W  /\  x  =  A )  ->  F/ y ps )
82, 4, 5, 7sbciedf 2874 . . 3  |-  ( ( B  e.  W  /\  x  =  A )  ->  ( [. B  / 
y ]. ph  <->  ps )
)
98adantll 460 . 2  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  x  =  A )  ->  ( [. B  /  y ]. ph  <->  ps ) )
10 nfv 1466 . . 3  |-  F/ x  A  e.  V
11 sbc2iegf.3 . . 3  |-  F/ x  B  e.  W
1210, 11nfan 1502 . 2  |-  F/ x
( A  e.  V  /\  B  e.  W
)
13 sbc2iegf.1 . . 3  |-  F/ x ps
1413a1i 9 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  F/ x ps )
151, 9, 12, 14sbciedf 2874 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( [. A  /  x ]. [. B  / 
y ]. ph  <->  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289   F/wnf 1394    e. wcel 1438   [.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:  sbc2ie  2910  opelopabaf  4100
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