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	Could not format assertion : No typesetting found for |- ( AE x e. A ( ph -> ps ) <-> AE x e. A ( ch -> th ) ) with typecode |-

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	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-rals	Could not format ( AE x e. A ( ch -> th ) <-> ( A. x e. A ( ch -> th ) /\ E. x e. A ch ) ) : No typesetting found for |- ( AE x e. A ( ch -> th ) <-> ( A. x e. A ( ch -> th ) /\ E. x e. A ch ) ) with typecode |-
9	6 7 8	3bitr4i	Could not format ( AE x e. A ( ph -> ps ) <-> AE x e. A ( ch -> th ) ) : No typesetting found for |- ( AE x e. A ( ph -> ps ) <-> AE x e. A ( ch -> th ) ) with typecode |-