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Theorem pm13.183 2733
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 2088 . 2  |-  ( y  =  A  ->  (
y  =  B  <->  A  =  B ) )
2 eqeq2 2091 . . . 4  |-  ( y  =  A  ->  (
z  =  y  <->  z  =  A ) )
32bibi1d 231 . . 3  |-  ( y  =  A  ->  (
( z  =  y  <-> 
z  =  B )  <-> 
( z  =  A  <-> 
z  =  B ) ) )
43albidv 1746 . 2  |-  ( y  =  A  ->  ( A. z ( z  =  y  <->  z  =  B )  <->  A. z ( z  =  A  <->  z  =  B ) ) )
5 eqeq2 2091 . . . 4  |-  ( y  =  B  ->  (
z  =  y  <->  z  =  B ) )
65alrimiv 1796 . . 3  |-  ( y  =  B  ->  A. z
( z  =  y  <-> 
z  =  B ) )
7 stdpc4 1699 . . . 4  |-  ( A. z ( z  =  y  <->  z  =  B )  ->  [ y  /  z ] ( z  =  y  <->  z  =  B ) )
8 sbbi 1875 . . . . 5  |-  ( [ y  /  z ] ( z  =  y  <-> 
z  =  B )  <-> 
( [ y  / 
z ] z  =  y  <->  [ y  /  z ] z  =  B ) )
9 eqsb3 2183 . . . . . . 7  |-  ( [ y  /  z ] z  =  B  <->  y  =  B )
109bibi2i 225 . . . . . 6  |-  ( ( [ y  /  z ] z  =  y  <->  [ y  /  z ] z  =  B )  <->  ( [ y  /  z ] z  =  y  <->  y  =  B ) )
11 equsb1 1709 . . . . . . 7  |-  [ y  /  z ] z  =  y
12 bi1 116 . . . . . . 7  |-  ( ( [ y  /  z ] z  =  y  <-> 
y  =  B )  ->  ( [ y  /  z ] z  =  y  ->  y  =  B ) )
1311, 12mpi 15 . . . . . 6  |-  ( ( [ y  /  z ] z  =  y  <-> 
y  =  B )  ->  y  =  B )
1410, 13sylbi 119 . . . . 5  |-  ( ( [ y  /  z ] z  =  y  <->  [ y  /  z ] z  =  B )  ->  y  =  B )
158, 14sylbi 119 . . . 4  |-  ( [ y  /  z ] ( z  =  y  <-> 
z  =  B )  ->  y  =  B )
167, 15syl 14 . . 3  |-  ( A. z ( z  =  y  <->  z  =  B )  ->  y  =  B )
176, 16impbii 124 . 2  |-  ( y  =  B  <->  A. z
( z  =  y  <-> 
z  =  B ) )
181, 4, 17vtoclbg 2660 1  |-  ( A  e.  V  ->  ( A  =  B  <->  A. z
( z  =  A  <-> 
z  =  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1283    = wceq 1285    e. wcel 1434   [wsb 1686
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604
This theorem is referenced by:  mpt22eqb  5641
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