Theorem bj-2albi 31782

Description: Closed form of 2albii 1738. (Contributed by BJ, 6-May-2019.)

Assertion

Ref	Expression
bj-2albi	⊢ (∀𝑥∀𝑦(𝜑 ↔ 𝜓) → (∀𝑥∀𝑦𝜑 ↔ ∀𝑥∀𝑦𝜓))

Proof of Theorem bj-2albi

Step	Hyp	Ref	Expression
1		albi 1736	. . 3 ⊢ (∀𝑦(𝜑 ↔ 𝜓) → (∀𝑦𝜑 ↔ ∀𝑦𝜓))
2	1	alimi 1730	. 2 ⊢ (∀𝑥∀𝑦(𝜑 ↔ 𝜓) → ∀𝑥(∀𝑦𝜑 ↔ ∀𝑦𝜓))
3		albi 1736	. 2 ⊢ (∀𝑥(∀𝑦𝜑 ↔ ∀𝑦𝜓) → (∀𝑥∀𝑦𝜑 ↔ ∀𝑥∀𝑦𝜓))
4	2, 3	syl 17	1 ⊢ (∀𝑥∀𝑦(𝜑 ↔ 𝜓) → (∀𝑥∀𝑦𝜑 ↔ ∀𝑥∀𝑦𝜓))

Syntax hints:   → wi 4   ↔ wb 195  ∀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

This theorem is referenced by: (None)