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Metamath Proof Explorer

Theorem pm14.123a

Description: Theorem *14.123 in WhiteheadRussell p. 185. (Contributed by Andrew Salmon, 9-Jun-2011)

Ref Expression
Assertion pm14.123a 
|- ( ( A e. V /\ B e. W ) -> ( A. z A. w ( ph <-> ( z = A /\ w = B ) ) <-> ( A. z A. w ( ph -> ( z = A /\ w = B ) ) /\ [. A / z ]. [. B / w ]. ph ) ) )

Proof

Step Hyp Ref Expression
1 2albiim 
 |-  ( A. z A. w ( ph <-> ( z = A /\ w = B ) ) <-> ( A. z A. w ( ph -> ( z = A /\ w = B ) ) /\ A. z A. w ( ( z = A /\ w = B ) -> ph ) ) )
2 2sbc6g 
 |-  ( ( A e. V /\ B e. W ) -> ( A. z A. w ( ( z = A /\ w = B ) -> ph ) <-> [. A / z ]. [. B / w ]. ph ) )
3 2 anbi2d 
 |-  ( ( A e. V /\ B e. W ) -> ( ( A. z A. w ( ph -> ( z = A /\ w = B ) ) /\ A. z A. w ( ( z = A /\ w = B ) -> ph ) ) <-> ( A. z A. w ( ph -> ( z = A /\ w = B ) ) /\ [. A / z ]. [. B / w ]. ph ) ) )
4 1 3 bitrid 
 |-  ( ( A e. V /\ B e. W ) -> ( A. z A. w ( ph <-> ( z = A /\ w = B ) ) <-> ( A. z A. w ( ph -> ( z = A /\ w = B ) ) /\ [. A / z ]. [. B / w ]. ph ) ) )