Theorem dfrals2

Description: The bounded "all some" form is the general form with the class membership folded into the antecedent. (Contributed by David A. Wheeler, 22-Oct-2018) (Revised by David A. Wheeler, 12-Jul-2026)

	Ref	Expression
Assertion	dfrals2	Could not format assertion : No typesetting found for |- ( AE x e. A ( ph -> ps ) <-> AE x ( ( x e. A /\ ph ) -> ps ) ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		df-ral	$$⊢\forall x\in A\left(\phi \to \psi \right)\leftrightarrow \forall x\left(x\in A\to \left(\phi \to \psi \right)\right)$$
2		impexp	$$⊢\left(\left(x\in A\wedge \phi \right)\to \psi \right)\leftrightarrow \left(x\in A\to \left(\phi \to \psi \right)\right)$$
3	2	albii	$$⊢\forall x\left(\left(x\in A\wedge \phi \right)\to \psi \right)\leftrightarrow \forall x\left(x\in A\to \left(\phi \to \psi \right)\right)$$
4	1 3	bitr4i	$$⊢\forall x\in A\left(\phi \to \psi \right)\leftrightarrow \forall x\left(\left(x\in A\wedge \phi \right)\to \psi \right)$$
5		df-rex	$$⊢\exists x\in A\phi \leftrightarrow \exists x\left(x\in A\wedge \phi \right)$$
6	4 5	anbi12i	$$⊢\left(\forall x\in A\left(\phi \to \psi \right)\wedge \exists x\in A\phi \right)\leftrightarrow \left(\forall x\left(\left(x\in A\wedge \phi \right)\to \psi \right)\wedge \exists x\left(x\in A\wedge \phi \right)\right)$$
7		df-rals	Could not format ( AE x e. A ( ph -> ps ) <-> ( A. x e. A ( ph -> ps ) /\ E. x e. A ph ) ) : No typesetting found for |- ( AE x e. A ( ph -> ps ) <-> ( A. x e. A ( ph -> ps ) /\ E. x e. A ph ) ) with typecode |-
8		df-als	Could not format ( AE x ( ( x e. A /\ ph ) -> ps ) <-> ( A. x ( ( x e. A /\ ph ) -> ps ) /\ E. x ( x e. A /\ ph ) ) ) : No typesetting found for |- ( AE x ( ( x e. A /\ ph ) -> ps ) <-> ( A. x ( ( x e. A /\ ph ) -> ps ) /\ E. x ( x e. A /\ ph ) ) ) with typecode |-
9	6 7 8	3bitr4i	Could not format ( AE x e. A ( ph -> ps ) <-> AE x ( ( x e. A /\ ph ) -> ps ) ) : No typesetting found for |- ( AE x e. A ( ph -> ps ) <-> AE x ( ( x e. A /\ ph ) -> ps ) ) with typecode |-