Theorem 2albi 44369

Description: Theorem *11.33 in [WhiteheadRussell] p. 162. Theorem 19.15 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)

Assertion

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

Proof of Theorem 2albi

Step	Hyp	Ref	Expression
1		albi 1817	. . 3 ⊢ (∀𝑦(𝜑 ↔ 𝜓) → (∀𝑦𝜑 ↔ ∀𝑦𝜓))
2	1	alimi 1810	. 2 ⊢ (∀𝑥∀𝑦(𝜑 ↔ 𝜓) → ∀𝑥(∀𝑦𝜑 ↔ ∀𝑦𝜓))
3		albi 1817	. 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 206  ∀wal 1537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808

This theorem depends on definitions:  df-bi 207

This theorem is referenced by: (None)