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Theorem sbeqalb 2871
Description: Theorem *14.121 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 28-Jun-2011.) (Proof shortened by Wolf Lammen, 9-May-2013.)
Assertion
Ref Expression
sbeqalb  |-  ( A  e.  V  ->  (
( A. x (
ph 
<->  x  =  A )  /\  A. x (
ph 
<->  x  =  B ) )  ->  A  =  B ) )
Distinct variable groups:    x, A    x, B
Allowed substitution hints:    ph( x)    V( x)

Proof of Theorem sbeqalb
StepHypRef Expression
1 bibi1 238 . . . . 5  |-  ( (
ph 
<->  x  =  A )  ->  ( ( ph  <->  x  =  B )  <->  ( x  =  A  <->  x  =  B
) ) )
21biimpa 290 . . . 4  |-  ( ( ( ph  <->  x  =  A )  /\  ( ph 
<->  x  =  B ) )  ->  ( x  =  A  <->  x  =  B
) )
32biimpd 142 . . 3  |-  ( ( ( ph  <->  x  =  A )  /\  ( ph 
<->  x  =  B ) )  ->  ( x  =  A  ->  x  =  B ) )
43alanimi 1389 . 2  |-  ( ( A. x ( ph  <->  x  =  A )  /\  A. x ( ph  <->  x  =  B ) )  ->  A. x ( x  =  A  ->  x  =  B ) )
5 sbceqal 2870 . 2  |-  ( A  e.  V  ->  ( A. x ( x  =  A  ->  x  =  B )  ->  A  =  B ) )
64, 5syl5 32 1  |-  ( A  e.  V  ->  (
( A. x (
ph 
<->  x  =  A )  /\  A. x (
ph 
<->  x  =  B ) )  ->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1283    = wceq 1285    e. wcel 1434
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  df-sbc 2817
This theorem is referenced by:  iotaval  4908
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