Theorem r19.30

Description: Restricted quantifier version of 19.30 . (Contributed by Scott Fenton, 25-Feb-2011) (Proof shortened by Wolf Lammen, 18-Jun-2023)

	Ref	Expression
Assertion	r19.30	$$⊢\forall x\in A\left(\phi \vee \psi \right)\to \left(\forall x\in A\phi \vee \exists x\in A\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		pm2.53	$$⊢\left(\psi \vee \phi \right)\to \left(\neg \psi \to \phi \right)$$
2	1	orcoms	$$⊢\left(\phi \vee \psi \right)\to \left(\neg \psi \to \phi \right)$$
3	2	ralimi	$$⊢\forall x\in A\left(\phi \vee \psi \right)\to \forall x\in A\left(\neg \psi \to \phi \right)$$
4		ralim	$$⊢\forall x\in A\left(\neg \psi \to \phi \right)\to \left(\forall x\in A\neg \psi \to \forall x\in A\phi \right)$$
5		ralnex	$$⊢\forall x\in A\neg \psi \leftrightarrow \neg \exists x\in A\psi$$
6	5	biimpri	$$⊢\neg \exists x\in A\psi \to \forall x\in A\neg \psi$$
7	6	imim1i	$$⊢\left(\forall x\in A\neg \psi \to \forall x\in A\phi \right)\to \left(\neg \exists x\in A\psi \to \forall x\in A\phi \right)$$
8	7	orrd	$$⊢\left(\forall x\in A\neg \psi \to \forall x\in A\phi \right)\to \left(\exists x\in A\psi \vee \forall x\in A\phi \right)$$
9	8	orcomd	$$⊢\left(\forall x\in A\neg \psi \to \forall x\in A\phi \right)\to \left(\forall x\in A\phi \vee \exists x\in A\psi \right)$$
10	3 4 9	3syl	$$⊢\forall x\in A\left(\phi \vee \psi \right)\to \left(\forall x\in A\phi \vee \exists x\in A\psi \right)$$