Theorem 2stdpc5

Description: A double stdpc5 (one direction of PM*11.3). See also 2stdpc4 and 19.21vv . (Contributed by BJ, 15-Sep-2018) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	2stdpc5.1	$$⊢Ⅎx\phi$$
	2stdpc5.2	$$⊢Ⅎy\phi$$
Assertion	2stdpc5	$$⊢\forall x\forall y\left(\phi \to \psi \right)\to \left(\phi \to \forall x\forall y\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		2stdpc5.1	$$⊢Ⅎx\phi$$
2		2stdpc5.2	$$⊢Ⅎy\phi$$
3	2	stdpc5	$$⊢\forall y\left(\phi \to \psi \right)\to \left(\phi \to \forall y\psi \right)$$
4	3	alimi	$$⊢\forall x\forall y\left(\phi \to \psi \right)\to \forall x\left(\phi \to \forall y\psi \right)$$
5	1	stdpc5	$$⊢\forall x\left(\phi \to \forall y\psi \right)\to \left(\phi \to \forall x\forall y\psi \right)$$
6	4 5	syl	$$⊢\forall x\forall y\left(\phi \to \psi \right)\to \left(\phi \to \forall x\forall y\psi \right)$$