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	syl5bb	|- ( ( 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 ) ) )