Theorem 2exbi 44399

Description: Theorem *11.341 in [WhiteheadRussell] p. 162. Theorem 19.18 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)

Assertion

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

Proof of Theorem 2exbi

Step	Hyp	Ref	Expression
1		exbi 1847	. . 3 ⊢ (∀𝑦(𝜑 ↔ 𝜓) → (∃𝑦𝜑 ↔ ∃𝑦𝜓))
2	1	alimi 1811	. 2 ⊢ (∀𝑥∀𝑦(𝜑 ↔ 𝜓) → ∀𝑥(∃𝑦𝜑 ↔ ∃𝑦𝜓))
3		exbi 1847	. 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 1538  ∃wex 1779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809

This theorem depends on definitions:  df-bi 207  df-ex 1780

This theorem is referenced by: (None)