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