Theorem wl-alanbii

Description: This theorem extends alanimi to a biconditional. Recurrent usage stacks up more quantifiers. (Contributed by Wolf Lammen, 4-Oct-2019)

	Ref	Expression
Hypothesis	wl-alanbii.1	⊢ ( 𝜑 ↔ ( 𝜓 ∧ 𝜒 ) )
Assertion	wl-alanbii	⊢ ( ∀ 𝑥 𝜑 ↔ ( ∀ 𝑥 𝜓 ∧ ∀ 𝑥 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		wl-alanbii.1	⊢ ( 𝜑 ↔ ( 𝜓 ∧ 𝜒 ) )
2	1	albii	⊢ ( ∀ 𝑥 𝜑 ↔ ∀ 𝑥 ( 𝜓 ∧ 𝜒 ) )
3		19.26	⊢ ( ∀ 𝑥 ( 𝜓 ∧ 𝜒 ) ↔ ( ∀ 𝑥 𝜓 ∧ ∀ 𝑥 𝜒 ) )
4	2 3	bitri	⊢ ( ∀ 𝑥 𝜑 ↔ ( ∀ 𝑥 𝜓 ∧ ∀ 𝑥 𝜒 ) )