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Theorem 2sbc5g 27285
Description: Theorem *13.22 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.)
Assertion
Ref Expression
2sbc5g  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( E. z E. 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 2sbc5g
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq2 2396 . . . . . . 7  |-  ( y  =  B  ->  (
w  =  y  <->  w  =  B ) )
21anbi2d 685 . . . . . 6  |-  ( y  =  B  ->  (
( z  =  x  /\  w  =  y )  <->  ( z  =  x  /\  w  =  B ) ) )
32anbi1d 686 . . . . 5  |-  ( y  =  B  ->  (
( ( z  =  x  /\  w  =  y )  /\  ph ) 
<->  ( ( z  =  x  /\  w  =  B )  /\  ph ) ) )
432exbidv 1635 . . . 4  |-  ( y  =  B  ->  ( E. z E. w ( ( z  =  x  /\  w  =  y )  /\  ph )  <->  E. z E. w ( ( z  =  x  /\  w  =  B )  /\  ph )
) )
5 dfsbcq 3106 . . . . 5  |-  ( y  =  B  ->  ( [. y  /  w ]. ph  <->  [. B  /  w ]. ph ) )
65sbcbidv 3158 . . . 4  |-  ( y  =  B  ->  ( [. x  /  z ]. [. y  /  w ]. ph  <->  [. x  /  z ]. [. B  /  w ]. ph ) )
74, 6bibi12d 313 . . 3  |-  ( y  =  B  ->  (
( E. z E. w ( ( z  =  x  /\  w  =  y )  /\  ph )  <->  [. x  /  z ]. [. y  /  w ]. ph )  <->  ( E. z E. w ( ( z  =  x  /\  w  =  B )  /\  ph )  <->  [. x  / 
z ]. [. B  /  w ]. ph ) ) )
8 eqeq2 2396 . . . . . . 7  |-  ( x  =  A  ->  (
z  =  x  <->  z  =  A ) )
98anbi1d 686 . . . . . 6  |-  ( x  =  A  ->  (
( z  =  x  /\  w  =  B )  <->  ( z  =  A  /\  w  =  B ) ) )
109anbi1d 686 . . . . 5  |-  ( x  =  A  ->  (
( ( z  =  x  /\  w  =  B )  /\  ph ) 
<->  ( ( z  =  A  /\  w  =  B )  /\  ph ) ) )
11102exbidv 1635 . . . 4  |-  ( x  =  A  ->  ( E. z E. w ( ( z  =  x  /\  w  =  B )  /\  ph )  <->  E. z E. w ( ( z  =  A  /\  w  =  B )  /\  ph )
) )
12 dfsbcq 3106 . . . 4  |-  ( x  =  A  ->  ( [. x  /  z ]. [. B  /  w ]. ph  <->  [. A  /  z ]. [. B  /  w ]. ph ) )
1311, 12bibi12d 313 . . 3  |-  ( x  =  A  ->  (
( E. z E. w ( ( z  =  x  /\  w  =  B )  /\  ph ) 
<-> 
[. x  /  z ]. [. B  /  w ]. ph )  <->  ( E. z E. w ( ( z  =  A  /\  w  =  B )  /\  ph )  <->  [. A  / 
z ]. [. B  /  w ]. ph ) ) )
14 sbc5 3128 . . . 4  |-  ( [. x  /  z ]. [. y  /  w ]. ph  <->  E. z
( z  =  x  /\  [. y  /  w ]. ph ) )
15 19.42v 1917 . . . . . 6  |-  ( E. w ( z  =  x  /\  ( w  =  y  /\  ph ) )  <->  ( z  =  x  /\  E. w
( w  =  y  /\  ph ) ) )
16 anass 631 . . . . . . 7  |-  ( ( ( z  =  x  /\  w  =  y )  /\  ph )  <->  ( z  =  x  /\  ( w  =  y  /\  ph ) ) )
1716exbii 1589 . . . . . 6  |-  ( E. w ( ( z  =  x  /\  w  =  y )  /\  ph )  <->  E. w ( z  =  x  /\  (
w  =  y  /\  ph ) ) )
18 sbc5 3128 . . . . . . 7  |-  ( [. y  /  w ]. ph  <->  E. w
( w  =  y  /\  ph ) )
1918anbi2i 676 . . . . . 6  |-  ( ( z  =  x  /\  [. y  /  w ]. ph )  <->  ( z  =  x  /\  E. w
( w  =  y  /\  ph ) ) )
2015, 17, 193bitr4ri 270 . . . . 5  |-  ( ( z  =  x  /\  [. y  /  w ]. ph )  <->  E. w ( ( z  =  x  /\  w  =  y )  /\  ph ) )
2120exbii 1589 . . . 4  |-  ( E. z ( z  =  x  /\  [. y  /  w ]. ph )  <->  E. z E. w ( ( z  =  x  /\  w  =  y )  /\  ph )
)
2214, 21bitr2i 242 . . 3  |-  ( E. z E. w ( ( z  =  x  /\  w  =  y )  /\  ph )  <->  [. x  /  z ]. [. y  /  w ]. ph )
237, 13, 22vtocl2g 2958 . 2  |-  ( ( B  e.  D  /\  A  e.  C )  ->  ( E. z E. w ( ( z  =  A  /\  w  =  B )  /\  ph ) 
<-> 
[. A  /  z ]. [. B  /  w ]. ph ) )
2423ancoms 440 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( E. z E. 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   E.wex 1547    = wceq 1649    e. wcel 1717   [.wsbc 3104
This theorem is referenced by:  pm14.123b  27295
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 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-v 2901  df-sbc 3105
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