Theorem pm14.122a

Description: Theorem *14.122 in WhiteheadRussell p. 185. (Contributed by Andrew Salmon, 9-Jun-2011)

	Ref	Expression
Assertion	pm14.122a	|- ( A e. V -> ( A. x ( ph <-> x = A ) <-> ( A. x ( ph -> x = A ) /\ [. A / x ]. ph ) ) )

Proof

Step	Hyp	Ref	Expression
1		albiim	|- ( A. x ( ph <-> x = A ) <-> ( A. x ( ph -> x = A ) /\ A. x ( x = A -> ph ) ) )
2		sbc6g	|- ( A e. V -> ( [. A / x ]. ph <-> A. x ( x = A -> ph ) ) )
3	2	bicomd	|- ( A e. V -> ( A. x ( x = A -> ph ) <-> [. A / x ]. ph ) )
4	3	anbi2d	|- ( A e. V -> ( ( A. x ( ph -> x = A ) /\ A. x ( x = A -> ph ) ) <-> ( A. x ( ph -> x = A ) /\ [. A / x ]. ph ) ) )
5	1 4	syl5bb	|- ( A e. V -> ( A. x ( ph <-> x = A ) <-> ( A. x ( ph -> x = A ) /\ [. A / x ]. ph ) ) )