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Theorem 2sbc6g 27483
Description: Theorem *13.21 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.)
Assertion
Ref Expression
2sbc6g  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A. z A. w ( ( z  =  A  /\  w  =  B )  ->  ph )  <->  [. A  /  z ]. [. B  /  w ]. ph ) )
Distinct variable groups:    z, w, A    w, B, z
Allowed substitution hints:    ph( z, w)    C( z, w)    D( z, w)

Proof of Theorem 2sbc6g
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq2 2413 . . . . . . 7  |-  ( y  =  B  ->  (
w  =  y  <->  w  =  B ) )
21anbi2d 685 . . . . . 6  |-  ( y  =  B  ->  (
( z  =  x  /\  w  =  y )  <->  ( z  =  x  /\  w  =  B ) ) )
32imbi1d 309 . . . . 5  |-  ( y  =  B  ->  (
( ( z  =  x  /\  w  =  y )  ->  ph )  <->  ( ( z  =  x  /\  w  =  B )  ->  ph ) ) )
432albidv 1634 . . . 4  |-  ( y  =  B  ->  ( A. z A. w ( ( z  =  x  /\  w  =  y )  ->  ph )  <->  A. z A. w ( ( z  =  x  /\  w  =  B )  ->  ph )
) )
5 dfsbcq 3123 . . . . 5  |-  ( y  =  B  ->  ( [. y  /  w ]. ph  <->  [. B  /  w ]. ph ) )
65sbcbidv 3175 . . . 4  |-  ( y  =  B  ->  ( [. x  /  z ]. [. y  /  w ]. ph  <->  [. x  /  z ]. [. B  /  w ]. ph ) )
74, 6bibi12d 313 . . 3  |-  ( y  =  B  ->  (
( A. z A. w ( ( z  =  x  /\  w  =  y )  ->  ph )  <->  [. x  /  z ]. [. y  /  w ]. ph )  <->  ( A. z A. w ( ( z  =  x  /\  w  =  B )  ->  ph )  <->  [. x  / 
z ]. [. B  /  w ]. ph ) ) )
8 eqeq2 2413 . . . . . . 7  |-  ( x  =  A  ->  (
z  =  x  <->  z  =  A ) )
98anbi1d 686 . . . . . 6  |-  ( x  =  A  ->  (
( z  =  x  /\  w  =  B )  <->  ( z  =  A  /\  w  =  B ) ) )
109imbi1d 309 . . . . 5  |-  ( x  =  A  ->  (
( ( z  =  x  /\  w  =  B )  ->  ph )  <->  ( ( z  =  A  /\  w  =  B )  ->  ph ) ) )
11102albidv 1634 . . . 4  |-  ( x  =  A  ->  ( A. z A. w ( ( z  =  x  /\  w  =  B )  ->  ph )  <->  A. z A. w ( ( z  =  A  /\  w  =  B )  ->  ph )
) )
12 dfsbcq 3123 . . . 4  |-  ( x  =  A  ->  ( [. x  /  z ]. [. B  /  w ]. ph  <->  [. A  /  z ]. [. B  /  w ]. ph ) )
1311, 12bibi12d 313 . . 3  |-  ( x  =  A  ->  (
( A. z A. w ( ( z  =  x  /\  w  =  B )  ->  ph )  <->  [. x  /  z ]. [. B  /  w ]. ph )  <->  ( A. z A. w ( ( z  =  A  /\  w  =  B )  ->  ph )  <->  [. A  /  z ]. [. B  /  w ]. ph ) ) )
14 vex 2919 . . . . 5  |-  x  e. 
_V
1514sbc6 3147 . . . 4  |-  ( [. x  /  z ]. [. y  /  w ]. ph  <->  A. z
( z  =  x  ->  [. y  /  w ]. ph ) )
16 19.21v 1909 . . . . . 6  |-  ( A. w ( z  =  x  ->  ( w  =  y  ->  ph )
)  <->  ( z  =  x  ->  A. w
( w  =  y  ->  ph ) ) )
17 impexp 434 . . . . . . 7  |-  ( ( ( z  =  x  /\  w  =  y )  ->  ph )  <->  ( z  =  x  ->  ( w  =  y  ->  ph )
) )
1817albii 1572 . . . . . 6  |-  ( A. w ( ( z  =  x  /\  w  =  y )  ->  ph )  <->  A. w ( z  =  x  ->  (
w  =  y  ->  ph ) ) )
19 vex 2919 . . . . . . . 8  |-  y  e. 
_V
2019sbc6 3147 . . . . . . 7  |-  ( [. y  /  w ]. ph  <->  A. w
( w  =  y  ->  ph ) )
2120imbi2i 304 . . . . . 6  |-  ( ( z  =  x  ->  [. y  /  w ]. ph )  <->  ( z  =  x  ->  A. w
( w  =  y  ->  ph ) ) )
2216, 18, 213bitr4ri 270 . . . . 5  |-  ( ( z  =  x  ->  [. y  /  w ]. ph )  <->  A. w
( ( z  =  x  /\  w  =  y )  ->  ph )
)
2322albii 1572 . . . 4  |-  ( A. z ( z  =  x  ->  [. y  /  w ]. ph )  <->  A. z A. w ( ( z  =  x  /\  w  =  y )  ->  ph ) )
2415, 23bitr2i 242 . . 3  |-  ( A. z A. w ( ( z  =  x  /\  w  =  y )  ->  ph )  <->  [. x  / 
z ]. [. y  /  w ]. ph )
257, 13, 24vtocl2g 2975 . 2  |-  ( ( B  e.  D  /\  A  e.  C )  ->  ( A. z A. w ( ( z  =  A  /\  w  =  B )  ->  ph )  <->  [. A  /  z ]. [. B  /  w ]. ph ) )
2625ancoms 440 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A. z A. w ( ( z  =  A  /\  w  =  B )  ->  ph )  <->  [. A  /  z ]. [. B  /  w ]. ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1546    = wceq 1649    e. wcel 1721   [.wsbc 3121
This theorem is referenced by:  pm14.123a  27493
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-v 2918  df-sbc 3122
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