Theorem 2alanimi 40697

Description: Removes two universal quantifiers from a statement. (Contributed by Andrew Salmon, 24-May-2011.)

Hypothesis

Ref	Expression
2alanimi.1	⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Assertion

Ref	Expression
2alanimi	⊢ ((∀𝑥∀𝑦𝜑 ∧ ∀𝑥∀𝑦𝜓) → ∀𝑥∀𝑦𝜒)

Proof of Theorem 2alanimi

Step	Hyp	Ref	Expression
1		2alanimi.1	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
2	1	alanimi 1813	. 2 ⊢ ((∀𝑦𝜑 ∧ ∀𝑦𝜓) → ∀𝑦𝜒)
3	2	alanimi 1813	1 ⊢ ((∀𝑥∀𝑦𝜑 ∧ ∀𝑥∀𝑦𝜓) → ∀𝑥∀𝑦𝜒)

Syntax hints:   → wi 4   ∧ wa 398  ∀wal 1531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806

This theorem depends on definitions:  df-bi 209  df-an 399

This theorem is referenced by: (None)