Theorem r19.30

Description: Restricted quantifier version of 19.30 . (Contributed by Scott Fenton, 25-Feb-2011) (Proof shortened by Wolf Lammen, 5-Nov-2024)

	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	$$⊢\left(\phi \vee \psi \right)\to \left(\neg \phi \to \psi \right)$$
2	1	ralimi	$$⊢\forall x\in A\left(\phi \vee \psi \right)\to \forall x\in A\left(\neg \phi \to \psi \right)$$
3		rexnal	$$⊢\exists x\in A\neg \phi \leftrightarrow \neg \forall x\in A\phi$$
4	3	biimpri	$$⊢\neg \forall x\in A\phi \to \exists x\in A\neg \phi$$
5		rexim	$$⊢\forall x\in A\left(\neg \phi \to \psi \right)\to \left(\exists x\in A\neg \phi \to \exists x\in A\psi \right)$$
6	2 4 5	syl2im	$$⊢\forall x\in A\left(\phi \vee \psi \right)\to \left(\neg \forall x\in A\phi \to \exists x\in A\psi \right)$$
7	6	orrd	$$⊢\forall x\in A\left(\phi \vee \psi \right)\to \left(\forall x\in A\phi \vee \exists x\in A\psi \right)$$