all professors are brilliant

∀𝑥 (professor(𝑥) → brilliant(𝑥))

the income of any banker is greater than the income of any bedder

∀𝑥𝑦 (banker(𝑥) ∧ bedder(𝑦) → income(𝑥) > income(𝑦))

every student has a supervisor

∀𝑥 (student(𝑥) → ∃𝑦 supervises(𝑦, 𝑥))

every student’s tutor is a member of the same college

∀𝑥𝑦 (student(𝑥) ∧ college(𝑦) ∧ member(𝑥, 𝑦) → member(tutor(𝑥), 𝑦))

there exist infinitely many Pythagorean triples

∀𝑥 ∃𝑖𝑗𝑘 (𝑖 > n ∧ 𝑖² + 𝑗² = 𝑘²)

infinitary de Morgan laws

• ¬(∀𝑥 𝐴) ≃ ∃𝑥 ¬𝐴
• ¬(∃𝑥 𝐴) ≃ ∀𝑥 ¬𝐴

pulling quantifiers through conjunction and disjunction (provided 𝑥 is not free in 𝐵)

• (∀𝑥 𝐴) ∧ 𝐵 ≃ ∀𝑥 (𝐴 ∧ 𝐵)
• (∀𝑥 𝐴) ∨ 𝐵 ≃ ∀𝑥 (𝐴 ∨ 𝐵)
• (∃𝑥 𝐴) ∧ 𝐵 ≃ ∃𝑥 (𝐴 ∧ 𝐵)
• (∃𝑥 𝐴) ∨ 𝐵 ≃ ∃𝑥 (𝐴 ∨ 𝐵)

distributive laws:

• (∀𝑥 𝐴) ∧ (∀𝑥 𝐵) ≃ ∀𝑥 (𝐴 ∧ 𝐵)
• (∃𝑥 𝐴) ∨ (∃𝑥 𝐵) ≃ ∃𝑥 (𝐴 ∨ 𝐵)

implication 𝐴 → 𝐵 as ¬𝐴 ∨ 𝐵 (provided 𝑥 is not free in 𝐵):

• (∀𝑥 𝐴) → 𝐵 ≃ ∃𝑥 (𝐴 → 𝐵)
• (∃𝑥 𝐴) → 𝐵 ≃ ∀𝑥 (𝐴 → 𝐵)

expansion: ∀ and ∃ as infinitary conjunction and disjunction:

• ∀𝑥 𝐴 ≃ (∀𝑥 𝐴) ∧ 𝐴[t/𝑥]
• ∃𝑥 𝐴 ≃ (∃𝑥 𝐴) ∨ 𝐴[t/𝑥]