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Theorem pm13.183 3207
Description: Compare theorem *13.183 in [WhiteheadRussell] p. 178. Only  A is required to be a set. (Contributed by Andrew Salmon, 3-Jun-2011.)
Assertion
Ref Expression
pm13.183  |-  ( A  e.  V  ->  ( A  =  B  <->  A. z
( z  =  A  <-> 
z  =  B ) ) )
Distinct variable groups:    z, A    z, B
Allowed substitution hint:    V( z)

Proof of Theorem pm13.183
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2458 . 2  |-  ( y  =  A  ->  (
y  =  B  <->  A  =  B ) )
2 eqeq2 2469 . . . 4  |-  ( y  =  A  ->  (
z  =  y  <->  z  =  A ) )
32bibi1d 319 . . 3  |-  ( y  =  A  ->  (
( z  =  y  <-> 
z  =  B )  <-> 
( z  =  A  <-> 
z  =  B ) ) )
43albidv 1680 . 2  |-  ( y  =  A  ->  ( A. z ( z  =  y  <->  z  =  B )  <->  A. z ( z  =  A  <->  z  =  B ) ) )
5 eqeq2 2469 . . . 4  |-  ( y  =  B  ->  (
z  =  y  <->  z  =  B ) )
65alrimiv 1686 . . 3  |-  ( y  =  B  ->  A. z
( z  =  y  <-> 
z  =  B ) )
7 stdpc4 2054 . . . 4  |-  ( A. z ( z  =  y  <->  z  =  B )  ->  [ y  /  z ] ( z  =  y  <->  z  =  B ) )
8 sbbi 2103 . . . . 5  |-  ( [ y  /  z ] ( z  =  y  <-> 
z  =  B )  <-> 
( [ y  / 
z ] z  =  y  <->  [ y  /  z ] z  =  B ) )
9 eqsb3 2574 . . . . . . 7  |-  ( [ y  /  z ] z  =  B  <->  y  =  B )
109bibi2i 313 . . . . . 6  |-  ( ( [ y  /  z ] z  =  y  <->  [ y  /  z ] z  =  B )  <->  ( [ y  /  z ] z  =  y  <->  y  =  B ) )
11 equsb1 2067 . . . . . . 7  |-  [ y  /  z ] z  =  y
12 bi1 186 . . . . . . 7  |-  ( ( [ y  /  z ] z  =  y  <-> 
y  =  B )  ->  ( [ y  /  z ] z  =  y  ->  y  =  B ) )
1311, 12mpi 17 . . . . . 6  |-  ( ( [ y  /  z ] z  =  y  <-> 
y  =  B )  ->  y  =  B )
1410, 13sylbi 195 . . . . 5  |-  ( ( [ y  /  z ] z  =  y  <->  [ y  /  z ] z  =  B )  ->  y  =  B )
158, 14sylbi 195 . . . 4  |-  ( [ y  /  z ] ( z  =  y  <-> 
z  =  B )  ->  y  =  B )
167, 15syl 16 . . 3  |-  ( A. z ( z  =  y  <->  z  =  B )  ->  y  =  B )
176, 16impbii 188 . 2  |-  ( y  =  B  <->  A. z
( z  =  y  <-> 
z  =  B ) )
181, 4, 17vtoclbg 3137 1  |-  ( A  e.  V  ->  ( A  =  B  <->  A. z
( z  =  A  <-> 
z  =  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1368    = wceq 1370   [wsb 1702    e. wcel 1758
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-v 3080
This theorem is referenced by:  mpt22eqb  6312
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