Theorem 2exbi

Description: Theorem *11.341 in WhiteheadRussell p. 162. Theorem 19.18 of Margaris p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011)

	Ref	Expression
Assertion	2exbi	$$⊢\forall x\forall y\left(\phi \leftrightarrow \psi \right)\to \left(\exists x\exists y\phi \leftrightarrow \exists x\exists y\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		exbi	$$⊢\forall y\left(\phi \leftrightarrow \psi \right)\to \left(\exists y\phi \leftrightarrow \exists y\psi \right)$$
2	1	alimi	$$⊢\forall x\forall y\left(\phi \leftrightarrow \psi \right)\to \forall x\left(\exists y\phi \leftrightarrow \exists y\psi \right)$$
3		exbi	$$⊢\forall x\left(\exists y\phi \leftrightarrow \exists y\psi \right)\to \left(\exists x\exists y\phi \leftrightarrow \exists x\exists y\psi \right)$$
4	2 3	syl	$$⊢\forall x\forall y\left(\phi \leftrightarrow \psi \right)\to \left(\exists x\exists y\phi \leftrightarrow \exists x\exists y\psi \right)$$