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Theorem pm13.183 2910
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
Dummy variable  y is distinct from all other variables.
Allowed substitution hint:    V( z)

Proof of Theorem pm13.183
StepHypRef Expression
1 eqeq1 2291 . 2  |-  ( y  =  A  ->  (
y  =  B  <->  A  =  B ) )
2 eqeq2 2294 . . . 4  |-  ( y  =  A  ->  (
z  =  y  <->  z  =  A ) )
32bibi1d 312 . . 3  |-  ( y  =  A  ->  (
( z  =  y  <-> 
z  =  B )  <-> 
( z  =  A  <-> 
z  =  B ) ) )
43albidv 1612 . 2  |-  ( y  =  A  ->  ( A. z ( z  =  y  <->  z  =  B )  <->  A. z ( z  =  A  <->  z  =  B ) ) )
5 eqeq2 2294 . . . 4  |-  ( y  =  B  ->  (
z  =  y  <->  z  =  B ) )
65alrimiv 1618 . . 3  |-  ( y  =  B  ->  A. z
( z  =  y  <-> 
z  =  B ) )
7 stdpc4 1970 . . . 4  |-  ( A. z ( z  =  y  <->  z  =  B )  ->  [ y  /  z ] ( z  =  y  <->  z  =  B ) )
8 sbbi 2011 . . . . 5  |-  ( [ y  /  z ] ( z  =  y  <-> 
z  =  B )  <-> 
( [ y  / 
z ] z  =  y  <->  [ y  /  z ] z  =  B ) )
9 eqsb3 2386 . . . . . . 7  |-  ( [ y  /  z ] z  =  B  <->  y  =  B )
109bibi2i 306 . . . . . 6  |-  ( ( [ y  /  z ] z  =  y  <->  [ y  /  z ] z  =  B )  <->  ( [ y  /  z ] z  =  y  <->  y  =  B ) )
11 equsb1 1980 . . . . . . 7  |-  [ y  /  z ] z  =  y
12 bi1 180 . . . . . . 7  |-  ( ( [ y  /  z ] z  =  y  <-> 
y  =  B )  ->  ( [ y  /  z ] z  =  y  ->  y  =  B ) )
1311, 12mpi 18 . . . . . 6  |-  ( ( [ y  /  z ] z  =  y  <-> 
y  =  B )  ->  y  =  B )
1410, 13sylbi 189 . . . . 5  |-  ( ( [ y  /  z ] z  =  y  <->  [ y  /  z ] z  =  B )  ->  y  =  B )
158, 14sylbi 189 . . . 4  |-  ( [ y  /  z ] ( z  =  y  <-> 
z  =  B )  ->  y  =  B )
167, 15syl 17 . . 3  |-  ( A. z ( z  =  y  <->  z  =  B )  ->  y  =  B )
176, 16impbii 182 . 2  |-  ( y  =  B  <->  A. z
( z  =  y  <-> 
z  =  B ) )
181, 4, 17vtoclbg 2846 1  |-  ( A  e.  V  ->  ( A  =  B  <->  A. z
( z  =  A  <-> 
z  =  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178   A.wal 1528    = wceq 1624   [wsb 1631    e. wcel 1685
This theorem is referenced by:  mpt22eqb  5915
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-v 2792
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