Theorem pm10.541 40706

Description: Theorem *10.541 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.)

Assertion

Ref	Expression
pm10.541	⊢ (∀𝑥(𝜑 → (𝜒 ∨ 𝜓)) ↔ (𝜒 ∨ ∀𝑥(𝜑 → 𝜓)))

Distinct variable group:   𝜒,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem pm10.541

Step	Hyp	Ref	Expression
1		bi2.04 391	. . . 4 ⊢ ((𝜑 → (¬ 𝜒 → 𝜓)) ↔ (¬ 𝜒 → (𝜑 → 𝜓)))
2	1	albii 1820	. . 3 ⊢ (∀𝑥(𝜑 → (¬ 𝜒 → 𝜓)) ↔ ∀𝑥(¬ 𝜒 → (𝜑 → 𝜓)))
3		19.21v 1940	. . 3 ⊢ (∀𝑥(¬ 𝜒 → (𝜑 → 𝜓)) ↔ (¬ 𝜒 → ∀𝑥(𝜑 → 𝜓)))
4	2, 3	bitri 277	. 2 ⊢ (∀𝑥(𝜑 → (¬ 𝜒 → 𝜓)) ↔ (¬ 𝜒 → ∀𝑥(𝜑 → 𝜓)))
5		df-or 844	. . . 4 ⊢ ((𝜒 ∨ 𝜓) ↔ (¬ 𝜒 → 𝜓))
6	5	imbi2i 338	. . 3 ⊢ ((𝜑 → (𝜒 ∨ 𝜓)) ↔ (𝜑 → (¬ 𝜒 → 𝜓)))
7	6	albii 1820	. 2 ⊢ (∀𝑥(𝜑 → (𝜒 ∨ 𝜓)) ↔ ∀𝑥(𝜑 → (¬ 𝜒 → 𝜓)))
8		df-or 844	. 2 ⊢ ((𝜒 ∨ ∀𝑥(𝜑 → 𝜓)) ↔ (¬ 𝜒 → ∀𝑥(𝜑 → 𝜓)))
9	4, 7, 8	3bitr4i 305	1 ⊢ (∀𝑥(𝜑 → (𝜒 ∨ 𝜓)) ↔ (𝜒 ∨ ∀𝑥(𝜑 → 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∨ wo 843  ∀wal 1535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911

This theorem depends on definitions:  df-bi 209  df-or 844  df-ex 1781

This theorem is referenced by: (None)