Theorem 19.31vv 45122

Description: Theorem *11.44 in [WhiteheadRussell] p. 163. Theorem 19.31 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)

Assertion

Ref	Expression
19.31vv	⊢ (∀𝑥∀𝑦(𝜑 ∨ 𝜓) ↔ (∀𝑥∀𝑦𝜑 ∨ 𝜓))

Distinct variable groups:   𝜓,𝑥   𝜓,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)

Proof of Theorem 19.31vv

Step	Hyp	Ref	Expression
1		19.31v 1970	. . 3 ⊢ (∀𝑦(𝜑 ∨ 𝜓) ↔ (∀𝑦𝜑 ∨ 𝜓))
2	1	albii 1848	. 2 ⊢ (∀𝑥∀𝑦(𝜑 ∨ 𝜓) ↔ ∀𝑥(∀𝑦𝜑 ∨ 𝜓))
3		19.31v 1970	. 2 ⊢ (∀𝑥(∀𝑦𝜑 ∨ 𝜓) ↔ (∀𝑥∀𝑦𝜑 ∨ 𝜓))
4	2, 3	bitri 278	1 ⊢ (∀𝑥∀𝑦(𝜑 ∨ 𝜓) ↔ (∀𝑥∀𝑦𝜑 ∨ 𝜓))

This proof depends on syntax axioms:   ↔ wb 209   ∨ wo 860  ∀wal 1567

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939

This proof depends on definitions:  df-bi 210  df-or 861  df-ex 1809

This theorem is used by: (None)