MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbc2iegf Structured version   Unicode version

Theorem sbc2iegf 3343
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 455 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  A  e.  V )
2 simpl 455 . . . 4  |-  ( ( B  e.  W  /\  x  =  A )  ->  B  e.  W )
3 sbc2iegf.4 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)
43adantll 712 . . . 4  |-  ( ( ( B  e.  W  /\  x  =  A
)  /\  y  =  B )  ->  ( ph 
<->  ps ) )
5 nfv 1728 . . . 4  |-  F/ y ( B  e.  W  /\  x  =  A
)
6 sbc2iegf.2 . . . . 5  |-  F/ y ps
76a1i 11 . . . 4  |-  ( ( B  e.  W  /\  x  =  A )  ->  F/ y ps )
82, 4, 5, 7sbciedf 3312 . . 3  |-  ( ( B  e.  W  /\  x  =  A )  ->  ( [. B  / 
y ]. ph  <->  ps )
)
98adantll 712 . 2  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  x  =  A )  ->  ( [. B  /  y ]. ph  <->  ps ) )
10 nfv 1728 . . 3  |-  F/ x  A  e.  V
11 sbc2iegf.3 . . 3  |-  F/ x  B  e.  W
1210, 11nfan 1956 . 2  |-  F/ x
( A  e.  V  /\  B  e.  W
)
13 sbc2iegf.1 . . 3  |-  F/ x ps
1413a1i 11 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  F/ x ps )
151, 9, 12, 14sbciedf 3312 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( [. A  /  x ]. [. B  / 
y ]. ph  <->  ps )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405   F/wnf 1637    e. wcel 1842   [.wsbc 3276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-v 3060  df-sbc 3277
This theorem is referenced by:  sbc2ie  3344  opelopabaf  4713  elmptrab  20618
  Copyright terms: Public domain W3C validator