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Theorem sbeqalb 2993
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 239 . . . . 5  |-  ( (
ph 
<->  x  =  A )  ->  ( ( ph  <->  x  =  B )  <->  ( x  =  A  <->  x  =  B
) ) )
21biimpa 294 . . . 4  |-  ( ( ( ph  <->  x  =  A )  /\  ( ph 
<->  x  =  B ) )  ->  ( x  =  A  <->  x  =  B
) )
32biimpd 143 . . 3  |-  ( ( ( ph  <->  x  =  A )  /\  ( ph 
<->  x  =  B ) )  ->  ( x  =  A  ->  x  =  B ) )
43alanimi 1439 . 2  |-  ( ( A. x ( ph  <->  x  =  A )  /\  A. x ( ph  <->  x  =  B ) )  ->  A. x ( x  =  A  ->  x  =  B ) )
5 sbceqal 2992 . 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 103    <-> wb 104   A.wal 1333    = wceq 1335    e. wcel 2128
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-sbc 2938
This theorem is referenced by:  iotaval  5148
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