Theorem ralseubii

Description: Congruence for "all some one" restricted to a class. This is the "all some one" counterpart of ralsbii . (Contributed by David A. Wheeler, 21-Jul-2026)

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

Proof

Step	Hyp	Ref	Expression
1		ralseubii.1	$$⊢\phi \leftrightarrow \chi$$
2		ralseubii.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	reubii	$$⊢\exists !x\in A\phi \leftrightarrow \exists !x\in A\chi$$
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-ralseu	$$⊢\forall \exists !x\in A\left(\phi \to \psi \right)\leftrightarrow \left(\forall x\in A\left(\phi \to \psi \right)\wedge \exists !x\in A\phi \right)$$
8		df-ralseu	$$⊢\forall \exists !x\in A\left(\chi \to \theta \right)\leftrightarrow \left(\forall x\in A\left(\chi \to \theta \right)\wedge \exists !x\in A\chi \right)$$
9	6 7 8	3bitr4i	$$⊢\forall \exists !x\in A\left(\phi \to \psi \right)\leftrightarrow \forall \exists !x\in A\left(\chi \to \theta \right)$$