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Metamath Proof Explorer

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 bitrid 
 |-  ( A e. V -> ( A. x ( ph <-> x = A ) <-> ( A. x ( ph -> x = A ) /\ [. A / x ]. ph ) ) )