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Theorem pm14.123b 29706
Description: Theorem *14.123 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
Assertion
Ref Expression
pm14.123b  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A. z A. w ( ph  ->  ( z  =  A  /\  w  =  B )
)  /\  [. A  / 
z ]. [. B  /  w ]. ph )  <->  ( A. z A. w ( ph  ->  ( z  =  A  /\  w  =  B ) )  /\  E. z E. w ph )
) )
Distinct variable groups:    w, A, z    w, B, z
Allowed substitution hints:    ph( z, w)    V( z, w)    W( z, w)

Proof of Theorem pm14.123b
StepHypRef Expression
1 2sbc5g 29696 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( E. z E. w ( ( z  =  A  /\  w  =  B )  /\  ph ) 
<-> 
[. A  /  z ]. [. B  /  w ]. ph ) )
21adantr 465 . . 3  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  A. z A. w ( ph  ->  ( z  =  A  /\  w  =  B )
) )  ->  ( E. z E. w ( ( z  =  A  /\  w  =  B )  /\  ph )  <->  [. A  /  z ]. [. B  /  w ]. ph ) )
3 nfa1 1831 . . . . 5  |-  F/ z A. z A. w
( ph  ->  ( z  =  A  /\  w  =  B ) )
4 nfa2 1879 . . . . . 6  |-  F/ w A. z A. w (
ph  ->  ( z  =  A  /\  w  =  B ) )
5 simpr 461 . . . . . . 7  |-  ( ( ( z  =  A  /\  w  =  B )  /\  ph )  ->  ph )
6 2sp 1801 . . . . . . . 8  |-  ( A. z A. w ( ph  ->  ( z  =  A  /\  w  =  B ) )  ->  ( ph  ->  ( z  =  A  /\  w  =  B ) ) )
76ancrd 554 . . . . . . 7  |-  ( A. z A. w ( ph  ->  ( z  =  A  /\  w  =  B ) )  ->  ( ph  ->  ( ( z  =  A  /\  w  =  B )  /\  ph ) ) )
85, 7impbid2 204 . . . . . 6  |-  ( A. z A. w ( ph  ->  ( z  =  A  /\  w  =  B ) )  ->  (
( ( z  =  A  /\  w  =  B )  /\  ph ) 
<-> 
ph ) )
94, 8exbid 1820 . . . . 5  |-  ( A. z A. w ( ph  ->  ( z  =  A  /\  w  =  B ) )  ->  ( E. w ( ( z  =  A  /\  w  =  B )  /\  ph ) 
<->  E. w ph )
)
103, 9exbid 1820 . . . 4  |-  ( A. z A. w ( ph  ->  ( z  =  A  /\  w  =  B ) )  ->  ( E. z E. w ( ( z  =  A  /\  w  =  B )  /\  ph )  <->  E. z E. w ph ) )
1110adantl 466 . . 3  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  A. z A. w ( ph  ->  ( z  =  A  /\  w  =  B )
) )  ->  ( E. z E. w ( ( z  =  A  /\  w  =  B )  /\  ph )  <->  E. z E. w ph ) )
122, 11bitr3d 255 . 2  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  A. z A. w ( ph  ->  ( z  =  A  /\  w  =  B )
) )  ->  ( [. A  /  z ]. [. B  /  w ]. ph  <->  E. z E. w ph ) )
1312pm5.32da 641 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A. z A. w ( ph  ->  ( z  =  A  /\  w  =  B )
)  /\  [. A  / 
z ]. [. B  /  w ]. ph )  <->  ( A. z A. w ( ph  ->  ( z  =  A  /\  w  =  B ) )  /\  E. z E. w ph )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1367    = wceq 1369   E.wex 1586    e. wcel 1756   [.wsbc 3207
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-v 2995  df-sbc 3208
This theorem is referenced by:  pm14.123c  29707
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