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Theorem 2sbc5g 36837
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 2482 . . . . . . 7  |-  ( y  =  B  ->  (
w  =  y  <->  w  =  B ) )
21anbi2d 718 . . . . . 6  |-  ( y  =  B  ->  (
( z  =  x  /\  w  =  y )  <->  ( z  =  x  /\  w  =  B ) ) )
32anbi1d 719 . . . . 5  |-  ( y  =  B  ->  (
( ( z  =  x  /\  w  =  y )  /\  ph ) 
<->  ( ( z  =  x  /\  w  =  B )  /\  ph ) ) )
432exbidv 1778 . . . 4  |-  ( y  =  B  ->  ( E. z E. w ( ( z  =  x  /\  w  =  y )  /\  ph )  <->  E. z E. w ( ( z  =  x  /\  w  =  B )  /\  ph )
) )
5 dfsbcq 3257 . . . . 5  |-  ( y  =  B  ->  ( [. y  /  w ]. ph  <->  [. B  /  w ]. ph ) )
65sbcbidv 3310 . . . 4  |-  ( y  =  B  ->  ( [. x  /  z ]. [. y  /  w ]. ph  <->  [. x  /  z ]. [. B  /  w ]. ph ) )
74, 6bibi12d 328 . . 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 2482 . . . . . . 7  |-  ( x  =  A  ->  (
z  =  x  <->  z  =  A ) )
98anbi1d 719 . . . . . 6  |-  ( x  =  A  ->  (
( z  =  x  /\  w  =  B )  <->  ( z  =  A  /\  w  =  B ) ) )
109anbi1d 719 . . . . 5  |-  ( x  =  A  ->  (
( ( z  =  x  /\  w  =  B )  /\  ph ) 
<->  ( ( z  =  A  /\  w  =  B )  /\  ph ) ) )
11102exbidv 1778 . . . 4  |-  ( x  =  A  ->  ( E. z E. w ( ( z  =  x  /\  w  =  B )  /\  ph )  <->  E. z E. w ( ( z  =  A  /\  w  =  B )  /\  ph )
) )
12 dfsbcq 3257 . . . 4  |-  ( x  =  A  ->  ( [. x  /  z ]. [. B  /  w ]. ph  <->  [. A  /  z ]. [. B  /  w ]. ph ) )
1311, 12bibi12d 328 . . 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 3280 . . . 4  |-  ( [. x  /  z ]. [. y  /  w ]. ph  <->  E. z
( z  =  x  /\  [. y  /  w ]. ph ) )
15 19.42v 1842 . . . . . 6  |-  ( E. w ( z  =  x  /\  ( w  =  y  /\  ph ) )  <->  ( z  =  x  /\  E. w
( w  =  y  /\  ph ) ) )
16 anass 661 . . . . . . 7  |-  ( ( ( z  =  x  /\  w  =  y )  /\  ph )  <->  ( z  =  x  /\  ( w  =  y  /\  ph ) ) )
1716exbii 1726 . . . . . 6  |-  ( E. w ( ( z  =  x  /\  w  =  y )  /\  ph )  <->  E. w ( z  =  x  /\  (
w  =  y  /\  ph ) ) )
18 sbc5 3280 . . . . . . 7  |-  ( [. y  /  w ]. ph  <->  E. w
( w  =  y  /\  ph ) )
1918anbi2i 708 . . . . . 6  |-  ( ( z  =  x  /\  [. y  /  w ]. ph )  <->  ( z  =  x  /\  E. w
( w  =  y  /\  ph ) ) )
2015, 17, 193bitr4ri 286 . . . . 5  |-  ( ( z  =  x  /\  [. y  /  w ]. ph )  <->  E. w ( ( z  =  x  /\  w  =  y )  /\  ph ) )
2120exbii 1726 . . . 4  |-  ( E. z ( z  =  x  /\  [. y  /  w ]. ph )  <->  E. z E. w ( ( z  =  x  /\  w  =  y )  /\  ph )
)
2214, 21bitr2i 258 . . 3  |-  ( E. z E. w ( ( z  =  x  /\  w  =  y )  /\  ph )  <->  [. x  /  z ]. [. y  /  w ]. ph )
237, 13, 22vtocl2g 3097 . 2  |-  ( ( B  e.  D  /\  A  e.  C )  ->  ( E. z E. w ( ( z  =  A  /\  w  =  B )  /\  ph ) 
<-> 
[. A  /  z ]. [. B  /  w ]. ph ) )
2423ancoms 460 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 setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452   E.wex 1671    e. wcel 1904   [.wsbc 3255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-v 3033  df-sbc 3256
This theorem is referenced by:  pm14.123b  36847
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