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 ) )