Theorem pm11.53

Description: Theorem *11.53 in WhiteheadRussell p. 164. See pm11.53v for a version requiring fewer axioms. (Contributed by Andrew Salmon, 24-May-2011)

	Ref	Expression
Assertion	pm11.53	$$⊢\forall x\forall y\left(\phi \to \psi \right)\leftrightarrow \left(\exists x\phi \to \forall y\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		19.21v	$$⊢\forall y\left(\phi \to \psi \right)\leftrightarrow \left(\phi \to \forall y\psi \right)$$
2	1	albii	$$⊢\forall x\forall y\left(\phi \to \psi \right)\leftrightarrow \forall x\left(\phi \to \forall y\psi \right)$$
3		nfv	$$⊢Ⅎx\psi$$
4	3	nfal	$$⊢Ⅎx\forall y\psi$$
5	4	19.23	$$⊢\forall x\left(\phi \to \forall y\psi \right)\leftrightarrow \left(\exists x\phi \to \forall y\psi \right)$$
6	2 5	bitri	$$⊢\forall x\forall y\left(\phi \to \psi \right)\leftrightarrow \left(\exists x\phi \to \forall y\psi \right)$$