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Metamath Proof Explorer

Theorem sbeqalb

Description: Theorem *14.121 in WhiteheadRussell p. 185. (Contributed by Andrew Salmon, 28-Jun-2011) (Proof shortened by Wolf Lammen, 9-May-2013)

Ref Expression
Assertion sbeqalb 
|- ( A e. V -> ( ( A. x ( ph <-> x = A ) /\ A. x ( ph <-> x = B ) ) -> A = B ) )

Proof

Step Hyp Ref Expression
1 bibi1 
 |-  ( ( ph <-> x = A ) -> ( ( ph <-> x = B ) <-> ( x = A <-> x = B ) ) )
2 1 biimpa 
 |-  ( ( ( ph <-> x = A ) /\ ( ph <-> x = B ) ) -> ( x = A <-> x = B ) )
3 2 biimpd 
 |-  ( ( ( ph <-> x = A ) /\ ( ph <-> x = B ) ) -> ( x = A -> x = B ) )
4 3 alanimi 
 |-  ( ( A. x ( ph <-> x = A ) /\ A. x ( ph <-> x = B ) ) -> A. x ( x = A -> x = B ) )
5 sbceqal 
 |-  ( A e. V -> ( A. x ( x = A -> x = B ) -> A = B ) )
6 4 5 syl5 
 |-  ( A e. V -> ( ( A. x ( ph <-> x = A ) /\ A. x ( ph <-> x = B ) ) -> A = B ) )