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Theorem sbeqalb 3056
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 317 . . . . 5  |-  ( (
ph 
<->  x  =  A )  ->  ( ( ph  <->  x  =  B )  <->  ( x  =  A  <->  x  =  B
) ) )
21biimpa 470 . . . 4  |-  ( ( ( ph  <->  x  =  A )  /\  ( ph 
<->  x  =  B ) )  ->  ( x  =  A  <->  x  =  B
) )
32biimpd 198 . . 3  |-  ( ( ( ph  <->  x  =  A )  /\  ( ph 
<->  x  =  B ) )  ->  ( x  =  A  ->  x  =  B ) )
43alanimi 1552 . 2  |-  ( ( A. x ( ph  <->  x  =  A )  /\  A. x ( ph  <->  x  =  B ) )  ->  A. x ( x  =  A  ->  x  =  B ) )
5 sbceqal 3055 . 2  |-  ( A  e.  V  ->  ( A. x ( x  =  A  ->  x  =  B )  ->  A  =  B ) )
64, 5syl5 28 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    <-> wb 176    /\ wa 358   A.wal 1530    = wceq 1632    e. wcel 1696
This theorem is referenced by:  iotaval  5246
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-sbc 3005
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