Theorem alsbid

Description: Deduction form of alsbii . (Contributed by David A. Wheeler, 12-Jul-2026)

	Ref	Expression
Hypotheses	alsbid.1	$$⊢Ⅎx\phi$$
	alsbid.2	$$⊢\phi \to \left(\psi \leftrightarrow \theta \right)$$
	alsbid.3	$$⊢\phi \to \left(\chi \leftrightarrow \tau \right)$$
Assertion	alsbid	Could not format assertion : No typesetting found for |- ( ph -> ( AE x ( ps -> ch ) <-> AE x ( th -> ta ) ) ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		alsbid.1	$$⊢Ⅎx\phi$$
2		alsbid.2	$$⊢\phi \to \left(\psi \leftrightarrow \theta \right)$$
3		alsbid.3	$$⊢\phi \to \left(\chi \leftrightarrow \tau \right)$$
4	2 3	imbi12d	$$⊢\phi \to \left(\left(\psi \to \chi \right)\leftrightarrow \left(\theta \to \tau \right)\right)$$
5	1 4	albid	$$⊢\phi \to \left(\forall x\left(\psi \to \chi \right)\leftrightarrow \forall x\left(\theta \to \tau \right)\right)$$
6	1 2	exbid	$$⊢\phi \to \left(\exists x\psi \leftrightarrow \exists x\theta \right)$$
7	5 6	anbi12d	$$⊢\phi \to \left(\left(\forall x\left(\psi \to \chi \right)\wedge \exists x\psi \right)\leftrightarrow \left(\forall x\left(\theta \to \tau \right)\wedge \exists x\theta \right)\right)$$
8		df-als	Could not format ( AE x ( ps -> ch ) <-> ( A. x ( ps -> ch ) /\ E. x ps ) ) : No typesetting found for |- ( AE x ( ps -> ch ) <-> ( A. x ( ps -> ch ) /\ E. x ps ) ) with typecode |-
9		df-als	Could not format ( AE x ( th -> ta ) <-> ( A. x ( th -> ta ) /\ E. x th ) ) : No typesetting found for |- ( AE x ( th -> ta ) <-> ( A. x ( th -> ta ) /\ E. x th ) ) with typecode |-
10	7 8 9	3bitr4g	Could not format ( ph -> ( AE x ( ps -> ch ) <-> AE x ( th -> ta ) ) ) : No typesetting found for |- ( ph -> ( AE x ( ps -> ch ) <-> AE x ( th -> ta ) ) ) with typecode |-