Metamath Proof Explorer

Theorem pm14.123c

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

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

Proof

Step Hyp Ref Expression
1 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 ) ) )
2 pm14.123b 
 |-  ( ( A e. V /\ B e. W ) -> ( ( A. z A. w ( ph -> ( z = A /\ w = B ) ) /\ [. A / z ]. [. B / w ]. ph ) <-> ( A. z A. w ( ph -> ( z = A /\ w = B ) ) /\ E. z E. w ph ) ) )
3 1 2 bitrd 
 |-  ( ( A e. V /\ B e. W ) -> ( A. z A. w ( ph <-> ( z = A /\ w = B ) ) <-> ( A. z A. w ( ph -> ( z = A /\ w = B ) ) /\ E. z E. w ph ) ) )