For All the Planets Graph

P(x)∧(∃y∈YQ(y))≡ ∃y∈Y(P(x)∧Q(y))P(x)∨(∃y∈YQ(y))≡ ∃y∈Y(P(x)∨Q(y)), provided that Y≠∅P(x)→(∃y∈YQ(y))≡ ∃y∈Y(P(x)→Q(y)), provided that Y≠∅P(x)↚(∃y∈YQ(y))≡ ∃y∈Y(P(x)↚Q(y))P(x)∧(∀y∈YQ(y))≡ ∀y∈Y(P(x)∧Q(y)), provided that Y≠∅P(x)∨(∀y∈YQ(y))≡ ∀y∈Y(P(x)∨Q(y))P(x)→(∀y∈YQ(y))≡ ∀y∈Y(P(x)→Q(y))P(x)↚(∀y∈YQ(y))≡ ∀y∈Y(P(x)↚Q(y)), provided that Y≠∅{displaystyle {begin{aligned}P(x)land (exists {y}{in }mathbf {Y} ,Q(y))&equiv exists {y}{in }mathbf {Y} ,(P(x)land Q(y))\P(x)lor (exists {y}{in }mathbf {Y} ,Q(y))&equiv exists {y}{in }mathbf {Y} ,(P(x)lor Q(y)),~mathrm {provided~that} ~mathbf {Y} neq emptyset \P(x)to (exists {y}{in }mathbf {Y} ,Q(y))&equiv exists {y}{in }mathbf {Y} ,(P(x)to Q(y)),~mathrm {provided~that} ~mathbf {Y} neq emptyset \P(x)nleftarrow (exists {y}{in }mathbf {Y} ,Q(y))&equiv exists {y}{in }mathbf {Y} ,(P(x)nleftarrow Q(y))\P(x)land (forall {y}{in }mathbf {Y} ,Q(y))&equiv forall {y}{in }mathbf {Y} ,(P(x)land Q(y)),~mathrm {provided~that} ~mathbf {Y} neq emptyset \P(x)lor (forall {y}{in }mathbf {Y} ,Q(y))&equiv forall {y}{in }mathbf {Y} ,(P(x)lor Q(y))\P(x)to (forall {y}{in }mathbf {Y} ,Q(y))&equiv forall {y}{in }mathbf {Y} ,(P(x)to Q(y))\P(x)nleftarrow (forall {y}{in }mathbf {Y} ,Q(y))&equiv forall {y}{in }mathbf {Y} ,(P(x)nleftarrow Q(y)),~mathrm {provided~that} ~mathbf {Y} neq emptyset end{aligned}}}