Theorem ralals

Description: If ph holds for every x in A , then the general "all some" quantifier with class membership as its antecedent reduces to the assertion that some x in A satisfies ph . See ralrals for the restricted counterpart. (Contributed by Peter Mazsa, 19-Dec-2018) (Revised by David A. Wheeler, 15-Jul-2026)

	Ref	Expression
Assertion	ralals	$$⊢\forall x\in A\phi \to \left(\forall \exists x\left(x\in A\to \phi \right)\leftrightarrow \exists x\in A\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		alsralrex	$$⊢\forall \exists x\left(x\in A\to \phi \right)\leftrightarrow \left(\forall x\in A\phi \wedge \exists x\in A\phi \right)$$
2		ibar	$$⊢\forall x\in A\phi \to \left(\exists x\in A\phi \leftrightarrow \left(\forall x\in A\phi \wedge \exists x\in A\phi \right)\right)$$
3	2	bicomd	$$⊢\forall x\in A\phi \to \left(\left(\forall x\in A\phi \wedge \exists x\in A\phi \right)\leftrightarrow \exists x\in A\phi \right)$$
4	1 3	bitrid	$$⊢\forall x\in A\phi \to \left(\forall \exists x\left(x\in A\to \phi \right)\leftrightarrow \exists x\in A\phi \right)$$