Theorem pm11.62

Description: Theorem *11.62 in WhiteheadRussell p. 166. Importation combined with the rearrangement with quantifiers. (Contributed by Andrew Salmon, 24-May-2011)

	Ref	Expression
Assertion	pm11.62	⊢ ( ∀ 𝑥 ∀ 𝑦 ( ( 𝜑 ∧ 𝜓 ) → 𝜒 ) ↔ ∀ 𝑥 ( 𝜑 → ∀ 𝑦 ( 𝜓 → 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		impexp	⊢ ( ( ( 𝜑 ∧ 𝜓 ) → 𝜒 ) ↔ ( 𝜑 → ( 𝜓 → 𝜒 ) ) )
2	1	albii	⊢ ( ∀ 𝑦 ( ( 𝜑 ∧ 𝜓 ) → 𝜒 ) ↔ ∀ 𝑦 ( 𝜑 → ( 𝜓 → 𝜒 ) ) )
3		19.21v	⊢ ( ∀ 𝑦 ( 𝜑 → ( 𝜓 → 𝜒 ) ) ↔ ( 𝜑 → ∀ 𝑦 ( 𝜓 → 𝜒 ) ) )
4	2 3	bitri	⊢ ( ∀ 𝑦 ( ( 𝜑 ∧ 𝜓 ) → 𝜒 ) ↔ ( 𝜑 → ∀ 𝑦 ( 𝜓 → 𝜒 ) ) )
5	4	albii	⊢ ( ∀ 𝑥 ∀ 𝑦 ( ( 𝜑 ∧ 𝜓 ) → 𝜒 ) ↔ ∀ 𝑥 ( 𝜑 → ∀ 𝑦 ( 𝜓 → 𝜒 ) ) )