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Theorem sbc2iegf 3057
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 443 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  A  e.  V )
2 simpl 443 . . . 4  |-  ( ( B  e.  W  /\  x  =  A )  ->  B  e.  W )
3 sbc2iegf.4 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)
43adantll 694 . . . 4  |-  ( ( ( B  e.  W  /\  x  =  A
)  /\  y  =  B )  ->  ( ph 
<->  ps ) )
5 nfv 1605 . . . 4  |-  F/ y ( B  e.  W  /\  x  =  A
)
6 sbc2iegf.2 . . . . 5  |-  F/ y ps
76a1i 10 . . . 4  |-  ( ( B  e.  W  /\  x  =  A )  ->  F/ y ps )
82, 4, 5, 7sbciedf 3026 . . 3  |-  ( ( B  e.  W  /\  x  =  A )  ->  ( [. B  / 
y ]. ph  <->  ps )
)
98adantll 694 . 2  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  x  =  A )  ->  ( [. B  /  y ]. ph  <->  ps ) )
10 nfv 1605 . . 3  |-  F/ x  A  e.  V
11 sbc2iegf.3 . . 3  |-  F/ x  B  e.  W
1210, 11nfan 1771 . 2  |-  F/ x
( A  e.  V  /\  B  e.  W
)
13 sbc2iegf.1 . . 3  |-  F/ x ps
1413a1i 10 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  F/ x ps )
151, 9, 12, 14sbciedf 3026 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( [. A  /  x ]. [. B  / 
y ]. ph  <->  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   F/wnf 1531    = wceq 1623    e. wcel 1684   [.wsbc 2991
This theorem is referenced by:  sbc2ie  3058  opelopabaf  4288  elmptrab  17522
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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