Theorem 2exbi 45131

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 1880	. . 3 ⊢ (∀𝑦(𝜑 ↔ 𝜓) → (∃𝑦𝜑 ↔ ∃𝑦𝜓))
2	1	alimi 1844	. 2 ⊢ (∀𝑥∀𝑦(𝜑 ↔ 𝜓) → ∀𝑥(∃𝑦𝜑 ↔ ∃𝑦𝜓))
3		exbi 1880	. 2 ⊢ (∀𝑥(∃𝑦𝜑 ↔ ∃𝑦𝜓) → (∃𝑥∃𝑦𝜑 ↔ ∃𝑥∃𝑦𝜓))
4	2, 3	syl 18	1 ⊢ (∀𝑥∀𝑦(𝜑 ↔ 𝜓) → (∃𝑥∃𝑦𝜑 ↔ ∃𝑥∃𝑦𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842

This proof depends on definitions:  df-bi 210  df-ex 1813

This theorem is used by: (None)