Theorem pm11.62 45145

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

Assertion

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

Distinct variable group:   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥, 𝑦)   𝜒(𝑥, 𝑦)

Proof of Theorem pm11.62

Step	Hyp	Ref	Expression
1		impexp 456	. . . 4 ⊢ (((𝜑 ∧ 𝜓) → 𝜒) ↔ (𝜑 → (𝜓 → 𝜒)))
2	1	albii 1852	. . 3 ⊢ (∀𝑦((𝜑 ∧ 𝜓) → 𝜒) ↔ ∀𝑦(𝜑 → (𝜓 → 𝜒)))
3		19.21v 1972	. . 3 ⊢ (∀𝑦(𝜑 → (𝜓 → 𝜒)) ↔ (𝜑 → ∀𝑦(𝜓 → 𝜒)))
4	2, 3	bitri 278	. 2 ⊢ (∀𝑦((𝜑 ∧ 𝜓) → 𝜒) ↔ (𝜑 → ∀𝑦(𝜓 → 𝜒)))
5	4	albii 1852	1 ⊢ (∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝜒) ↔ ∀𝑥(𝜑 → ∀𝑦(𝜓 → 𝜒)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813

This theorem is used by: (None)