Theorem 19.31vv 42002

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 1944	. . 3 ⊢ (∀𝑦(𝜑 ∨ 𝜓) ↔ (∀𝑦𝜑 ∨ 𝜓))
2	1	albii 1822	. 2 ⊢ (∀𝑥∀𝑦(𝜑 ∨ 𝜓) ↔ ∀𝑥(∀𝑦𝜑 ∨ 𝜓))
3		19.31v 1944	. 2 ⊢ (∀𝑥(∀𝑦𝜑 ∨ 𝜓) ↔ (∀𝑥∀𝑦𝜑 ∨ 𝜓))
4	2, 3	bitri 274	1 ⊢ (∀𝑥∀𝑦(𝜑 ∨ 𝜓) ↔ (∀𝑥∀𝑦𝜑 ∨ 𝜓))

Syntax hints:   ↔ wb 205   ∨ wo 844  ∀wal 1537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913

This theorem depends on definitions:  df-bi 206  df-or 845  df-ex 1783

This theorem is referenced by: (None)