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Theorem pm14.123b 27536
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 27526 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( E. z E. w ( ( z  =  A  /\  w  =  B )  /\  ph ) 
<-> 
[. A  /  z ]. [. B  /  w ]. ph ) )
21adantr 452 . . 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 1806 . . . . 5  |-  F/ z A. z A. w
( ph  ->  ( z  =  A  /\  w  =  B ) )
4 nfa2 1874 . . . . . 6  |-  F/ w A. z A. w (
ph  ->  ( z  =  A  /\  w  =  B ) )
5 simpr 448 . . . . . . 7  |-  ( ( ( z  =  A  /\  w  =  B )  /\  ph )  ->  ph )
6 sp 1763 . . . . . . . . 9  |-  ( A. w ( ph  ->  ( z  =  A  /\  w  =  B )
)  ->  ( ph  ->  ( z  =  A  /\  w  =  B ) ) )
76sps 1770 . . . . . . . 8  |-  ( A. z A. w ( ph  ->  ( z  =  A  /\  w  =  B ) )  ->  ( ph  ->  ( z  =  A  /\  w  =  B ) ) )
87ancrd 538 . . . . . . 7  |-  ( A. z A. w ( ph  ->  ( z  =  A  /\  w  =  B ) )  ->  ( ph  ->  ( ( z  =  A  /\  w  =  B )  /\  ph ) ) )
95, 8impbid2 196 . . . . . 6  |-  ( A. z A. w ( ph  ->  ( z  =  A  /\  w  =  B ) )  ->  (
( ( z  =  A  /\  w  =  B )  /\  ph ) 
<-> 
ph ) )
104, 9exbid 1789 . . . . 5  |-  ( A. z A. w ( ph  ->  ( z  =  A  /\  w  =  B ) )  ->  ( E. w ( ( z  =  A  /\  w  =  B )  /\  ph ) 
<->  E. w ph )
)
113, 10exbid 1789 . . . 4  |-  ( A. z A. w ( ph  ->  ( z  =  A  /\  w  =  B ) )  ->  ( E. z E. w ( ( z  =  A  /\  w  =  B )  /\  ph )  <->  E. z E. w ph ) )
1211adantl 453 . . 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 ) )
132, 12bitr3d 247 . 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 ) )
1413pm5.32da 623 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1549   E.wex 1550    = wceq 1652    e. wcel 1725   [.wsbc 3148
This theorem is referenced by:  pm14.123c  27537
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-v 2945  df-sbc 3149
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