Theorem 2albi

Description: Theorem *11.33 in WhiteheadRussell p. 162. Theorem 19.15 of Margaris p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011)

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

Proof

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