Theorem alsbid

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

	Ref	Expression
Hypotheses	alsbid.1	⊢ Ⅎ 𝑥 𝜑
	alsbid.2	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜃 ) )
	alsbid.3	⊢ ( 𝜑 → ( 𝜒 ↔ 𝜏 ) )
Assertion	alsbid	⊢ ( 𝜑 → ( ∀∃ 𝑥 ( 𝜓 → 𝜒 ) ↔ ∀∃ 𝑥 ( 𝜃 → 𝜏 ) ) )

Proof

Step	Hyp	Ref	Expression
1		alsbid.1	⊢ Ⅎ 𝑥 𝜑
2		alsbid.2	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜃 ) )
3		alsbid.3	⊢ ( 𝜑 → ( 𝜒 ↔ 𝜏 ) )
4	2 3	imbi12d	⊢ ( 𝜑 → ( ( 𝜓 → 𝜒 ) ↔ ( 𝜃 → 𝜏 ) ) )
5	1 4	albid	⊢ ( 𝜑 → ( ∀ 𝑥 ( 𝜓 → 𝜒 ) ↔ ∀ 𝑥 ( 𝜃 → 𝜏 ) ) )
6	1 2	exbid	⊢ ( 𝜑 → ( ∃ 𝑥 𝜓 ↔ ∃ 𝑥 𝜃 ) )
7	5 6	anbi12d	⊢ ( 𝜑 → ( ( ∀ 𝑥 ( 𝜓 → 𝜒 ) ∧ ∃ 𝑥 𝜓 ) ↔ ( ∀ 𝑥 ( 𝜃 → 𝜏 ) ∧ ∃ 𝑥 𝜃 ) ) )
8		df-als	⊢ ( ∀∃ 𝑥 ( 𝜓 → 𝜒 ) ↔ ( ∀ 𝑥 ( 𝜓 → 𝜒 ) ∧ ∃ 𝑥 𝜓 ) )
9		df-als	⊢ ( ∀∃ 𝑥 ( 𝜃 → 𝜏 ) ↔ ( ∀ 𝑥 ( 𝜃 → 𝜏 ) ∧ ∃ 𝑥 𝜃 ) )
10	7 8 9	3bitr4g	⊢ ( 𝜑 → ( ∀∃ 𝑥 ( 𝜓 → 𝜒 ) ↔ ∀∃ 𝑥 ( 𝜃 → 𝜏 ) ) )