Theorem wl-alanbii 32530

Description: This theorem extends alanimi 1734 to a biconditional. Recurrent usage stacks up more quantifiers. (Contributed by Wolf Lammen, 4-Oct-2019.)

Hypothesis

Ref	Expression
wl-alanbii.1	⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))

Assertion

Ref	Expression
wl-alanbii	⊢ (∀𝑥𝜑 ↔ (∀𝑥𝜓 ∧ ∀𝑥𝜒))

Proof of Theorem wl-alanbii

Step	Hyp	Ref	Expression
1		wl-alanbii.1	. . 3 ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))
2	1	albii 1737	. 2 ⊢ (∀𝑥𝜑 ↔ ∀𝑥(𝜓 ∧ 𝜒))
3		19.26 1786	. 2 ⊢ (∀𝑥(𝜓 ∧ 𝜒) ↔ (∀𝑥𝜓 ∧ ∀𝑥𝜒))
4	2, 3	bitri 263	1 ⊢ (∀𝑥𝜑 ↔ (∀𝑥𝜓 ∧ ∀𝑥𝜒))

Syntax hints:   ↔ wb 195   ∧ wa 383  ∀wal 1473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728

This theorem depends on definitions:  df-bi 196  df-an 385

This theorem is referenced by: (None)