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Theorem pm14.123a 26994
Description: Theorem *14.123 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
Assertion
Ref Expression
pm14.123a  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A. z A. w ( ph  <->  ( z  =  A  /\  w  =  B ) )  <->  ( A. z A. w ( ph  ->  ( z  =  A  /\  w  =  B ) )  /\  [. A  /  z ]. [. B  /  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 pm14.123a
StepHypRef Expression
1 2albiim 1613 . 2  |-  ( A. z A. w ( ph  <->  ( z  =  A  /\  w  =  B )
)  <->  ( A. z A. w ( ph  ->  ( z  =  A  /\  w  =  B )
)  /\  A. z A. w ( ( z  =  A  /\  w  =  B )  ->  ph )
) )
2 2sbc6g 26984 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A. z A. w ( ( z  =  A  /\  w  =  B )  ->  ph )  <->  [. A  /  z ]. [. B  /  w ]. ph ) )
32anbi2d 687 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A. z A. w ( ph  ->  ( z  =  A  /\  w  =  B )
)  /\  A. z A. w ( ( z  =  A  /\  w  =  B )  ->  ph )
)  <->  ( A. z A. w ( ph  ->  ( z  =  A  /\  w  =  B )
)  /\  [. A  / 
z ]. [. B  /  w ]. ph ) ) )
41, 3syl5bb 250 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A. z A. w ( ph  <->  ( z  =  A  /\  w  =  B ) )  <->  ( A. z A. w ( ph  ->  ( z  =  A  /\  w  =  B ) )  /\  [. A  /  z ]. [. B  /  w ]. ph )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360   A.wal 1532    = wceq 1619    e. wcel 1621   [.wsbc 2966
This theorem is referenced by:  pm14.123c  26996
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 1927  ax-ext 2239
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 1884  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-v 2765  df-sbc 2967
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