Theorem pm11.62 41901

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 450	. . . 4 ⊢ (((𝜑 ∧ 𝜓) → 𝜒) ↔ (𝜑 → (𝜓 → 𝜒)))
2	1	albii 1823	. . 3 ⊢ (∀𝑦((𝜑 ∧ 𝜓) → 𝜒) ↔ ∀𝑦(𝜑 → (𝜓 → 𝜒)))
3		19.21v 1943	. . 3 ⊢ (∀𝑦(𝜑 → (𝜓 → 𝜒)) ↔ (𝜑 → ∀𝑦(𝜓 → 𝜒)))
4	2, 3	bitri 274	. 2 ⊢ (∀𝑦((𝜑 ∧ 𝜓) → 𝜒) ↔ (𝜑 → ∀𝑦(𝜓 → 𝜒)))
5	4	albii 1823	1 ⊢ (∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝜒) ↔ ∀𝑥(𝜑 → ∀𝑦(𝜓 → 𝜒)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784

This theorem is referenced by: (None)