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Theorem sbeqalb 3205
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 318 . . . . 5  |-  ( (
ph 
<->  x  =  A )  ->  ( ( ph  <->  x  =  B )  <->  ( x  =  A  <->  x  =  B
) ) )
21biimpa 471 . . . 4  |-  ( ( ( ph  <->  x  =  A )  /\  ( ph 
<->  x  =  B ) )  ->  ( x  =  A  <->  x  =  B
) )
32biimpd 199 . . 3  |-  ( ( ( ph  <->  x  =  A )  /\  ( ph 
<->  x  =  B ) )  ->  ( x  =  A  ->  x  =  B ) )
43alanimi 1571 . 2  |-  ( ( A. x ( ph  <->  x  =  A )  /\  A. x ( ph  <->  x  =  B ) )  ->  A. x ( x  =  A  ->  x  =  B ) )
5 sbceqal 3204 . 2  |-  ( A  e.  V  ->  ( A. x ( x  =  A  ->  x  =  B )  ->  A  =  B ) )
64, 5syl5 30 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 177    /\ wa 359   A.wal 1549    = wceq 1652    e. wcel 1725
This theorem is referenced by:  iotaval  5421
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  df-sbc 3154
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