MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pm13.183 Structured version   Unicode version

Theorem pm13.183 3068
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 2441 . 2  |-  ( y  =  A  ->  (
y  =  B  <->  A  =  B ) )
2 eqeq2 2444 . . . 4  |-  ( y  =  A  ->  (
z  =  y  <->  z  =  A ) )
32bibi1d 311 . . 3  |-  ( y  =  A  ->  (
( z  =  y  <-> 
z  =  B )  <-> 
( z  =  A  <-> 
z  =  B ) ) )
43albidv 1635 . 2  |-  ( y  =  A  ->  ( A. z ( z  =  y  <->  z  =  B )  <->  A. z ( z  =  A  <->  z  =  B ) ) )
5 eqeq2 2444 . . . 4  |-  ( y  =  B  ->  (
z  =  y  <->  z  =  B ) )
65alrimiv 1641 . . 3  |-  ( y  =  B  ->  A. z
( z  =  y  <-> 
z  =  B ) )
7 stdpc4 2087 . . . 4  |-  ( A. z ( z  =  y  <->  z  =  B )  ->  [ y  /  z ] ( z  =  y  <->  z  =  B ) )
8 sbbi 2145 . . . . 5  |-  ( [ y  /  z ] ( z  =  y  <-> 
z  =  B )  <-> 
( [ y  / 
z ] z  =  y  <->  [ y  /  z ] z  =  B ) )
9 eqsb3 2536 . . . . . . 7  |-  ( [ y  /  z ] z  =  B  <->  y  =  B )
109bibi2i 305 . . . . . 6  |-  ( ( [ y  /  z ] z  =  y  <->  [ y  /  z ] z  =  B )  <->  ( [ y  /  z ] z  =  y  <->  y  =  B ) )
11 equsb1 2113 . . . . . . 7  |-  [ y  /  z ] z  =  y
12 bi1 179 . . . . . . 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 188 . . . . 5  |-  ( ( [ y  /  z ] z  =  y  <->  [ y  /  z ] z  =  B )  ->  y  =  B )
158, 14sylbi 188 . . . 4  |-  ( [ y  /  z ] ( z  =  y  <-> 
z  =  B )  ->  y  =  B )
167, 15syl 16 . . 3  |-  ( A. z ( z  =  y  <->  z  =  B )  ->  y  =  B )
176, 16impbii 181 . 2  |-  ( y  =  B  <->  A. z
( z  =  y  <-> 
z  =  B ) )
181, 4, 17vtoclbg 3004 1  |-  ( A  e.  V  ->  ( A  =  B  <->  A. z
( z  =  A  <-> 
z  =  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1549    = wceq 1652   [wsb 1658    e. wcel 1725
This theorem is referenced by:  mpt22eqb  6171
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950
  Copyright terms: Public domain W3C validator