Theorem ralsbii

Description: Congruence for "all some" restricted to a class. (Contributed by David A. Wheeler, 12-Jul-2026)

	Ref	Expression
Hypotheses	ralsbii.1	$$⊢\phi \leftrightarrow \chi$$
	ralsbii.2	$$⊢\psi \leftrightarrow \theta$$
Assertion	ralsbii	$$⊢\forall \exists x\in A\left(\phi \to \psi \right)\leftrightarrow \forall \exists x\in A\left(\chi \to \theta \right)$$

Proof

Step	Hyp	Ref	Expression
1		ralsbii.1	$$⊢\phi \leftrightarrow \chi$$
2		ralsbii.2	$$⊢\psi \leftrightarrow \theta$$
3	1 2	imbi12i	$$⊢\left(\phi \to \psi \right)\leftrightarrow \left(\chi \to \theta \right)$$
4	3	ralbii	$$⊢\forall x\in A\left(\phi \to \psi \right)\leftrightarrow \forall x\in A\left(\chi \to \theta \right)$$
5	1	rexbii	$$⊢\exists x\in A\phi \leftrightarrow \exists x\in A\chi$$
6	4 5	anbi12i	$$⊢\left(\forall x\in A\left(\phi \to \psi \right)\wedge \exists x\in A\phi \right)\leftrightarrow \left(\forall x\in A\left(\chi \to \theta \right)\wedge \exists x\in A\chi \right)$$
7		df-rals	$$⊢\forall \exists x\in A\left(\phi \to \psi \right)\leftrightarrow \left(\forall x\in A\left(\phi \to \psi \right)\wedge \exists x\in A\phi \right)$$
8		df-rals	$$⊢\forall \exists x\in A\left(\chi \to \theta \right)\leftrightarrow \left(\forall x\in A\left(\chi \to \theta \right)\wedge \exists x\in A\chi \right)$$
9	6 7 8	3bitr4i	$$⊢\forall \exists x\in A\left(\phi \to \psi \right)\leftrightarrow \forall \exists x\in A\left(\chi \to \theta \right)$$