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Theorem 2sbc5g 26783
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
StepHypRef Expression
1 eqeq2 2262 . . . . . . 7  |-  ( y  =  B  ->  (
w  =  y  <->  w  =  B ) )
21anbi2d 687 . . . . . 6  |-  ( y  =  B  ->  (
( z  =  x  /\  w  =  y )  <->  ( z  =  x  /\  w  =  B ) ) )
32anbi1d 688 . . . . 5  |-  ( y  =  B  ->  (
( ( z  =  x  /\  w  =  y )  /\  ph ) 
<->  ( ( z  =  x  /\  w  =  B )  /\  ph ) ) )
432exbidv 2007 . . . 4  |-  ( y  =  B  ->  ( E. z E. w ( ( z  =  x  /\  w  =  y )  /\  ph )  <->  E. z E. w ( ( z  =  x  /\  w  =  B )  /\  ph )
) )
5 dfsbcq 2923 . . . . 5  |-  ( y  =  B  ->  ( [. y  /  w ]. ph  <->  [. B  /  w ]. ph ) )
65sbcbidv 2975 . . . 4  |-  ( y  =  B  ->  ( [. x  /  z ]. [. y  /  w ]. ph  <->  [. x  /  z ]. [. B  /  w ]. ph ) )
74, 6bibi12d 314 . . 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 2262 . . . . . . 7  |-  ( x  =  A  ->  (
z  =  x  <->  z  =  A ) )
98anbi1d 688 . . . . . 6  |-  ( x  =  A  ->  (
( z  =  x  /\  w  =  B )  <->  ( z  =  A  /\  w  =  B ) ) )
109anbi1d 688 . . . . 5  |-  ( x  =  A  ->  (
( ( z  =  x  /\  w  =  B )  /\  ph ) 
<->  ( ( z  =  A  /\  w  =  B )  /\  ph ) ) )
11102exbidv 2007 . . . 4  |-  ( x  =  A  ->  ( E. z E. w ( ( z  =  x  /\  w  =  B )  /\  ph )  <->  E. z E. w ( ( z  =  A  /\  w  =  B )  /\  ph )
) )
12 dfsbcq 2923 . . . 4  |-  ( x  =  A  ->  ( [. x  /  z ]. [. B  /  w ]. ph  <->  [. A  /  z ]. [. B  /  w ]. ph ) )
1311, 12bibi12d 314 . . 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 2945 . . . 4  |-  ( [. x  /  z ]. [. y  /  w ]. ph  <->  E. z
( z  =  x  /\  [. y  /  w ]. ph ) )
15 19.42v 2038 . . . . . 6  |-  ( E. w ( z  =  x  /\  ( w  =  y  /\  ph ) )  <->  ( z  =  x  /\  E. w
( w  =  y  /\  ph ) ) )
16 anass 633 . . . . . . 7  |-  ( ( ( z  =  x  /\  w  =  y )  /\  ph )  <->  ( z  =  x  /\  ( w  =  y  /\  ph ) ) )
1716exbii 1580 . . . . . 6  |-  ( E. w ( ( z  =  x  /\  w  =  y )  /\  ph )  <->  E. w ( z  =  x  /\  (
w  =  y  /\  ph ) ) )
18 sbc5 2945 . . . . . . 7  |-  ( [. y  /  w ]. ph  <->  E. w
( w  =  y  /\  ph ) )
1918anbi2i 678 . . . . . 6  |-  ( ( z  =  x  /\  [. y  /  w ]. ph )  <->  ( z  =  x  /\  E. w
( w  =  y  /\  ph ) ) )
2015, 17, 193bitr4ri 271 . . . . 5  |-  ( ( z  =  x  /\  [. y  /  w ]. ph )  <->  E. w ( ( z  =  x  /\  w  =  y )  /\  ph ) )
2120exbii 1580 . . . 4  |-  ( E. z ( z  =  x  /\  [. y  /  w ]. ph )  <->  E. z E. w ( ( z  =  x  /\  w  =  y )  /\  ph )
)
2214, 21bitr2i 243 . . 3  |-  ( E. z E. w ( ( z  =  x  /\  w  =  y )  /\  ph )  <->  [. x  /  z ]. [. y  /  w ]. ph )
237, 13, 22vtocl2g 2785 . 2  |-  ( ( B  e.  D  /\  A  e.  C )  ->  ( E. z E. w ( ( z  =  A  /\  w  =  B )  /\  ph ) 
<-> 
[. A  /  z ]. [. B  /  w ]. ph ) )
2423ancoms 441 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 6    <-> wb 178    /\ wa 360   E.wex 1537    = wceq 1619    e. wcel 1621   [.wsbc 2921
This theorem is referenced by:  pm14.123b  26793
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-v 2729  df-sbc 2922
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