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Theorem pm14.123a 26948
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 1612 . 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 26938 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A. z A. w ( ( z  =  A  /\  w  =  B )  ->  ph )  <->  [. A  /  z ]. [. B  /  w ]. ph ) )
32anbi2d 684 . 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 248 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 4    <-> wb 176    /\ wa 358   A.wal 1540    = wceq 1642    e. wcel 1710   [.wsbc 3067
This theorem is referenced by:  pm14.123c  26950
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-v 2866  df-sbc 3068
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