Theorem 19.32 2225

Description: Theorem 19.32 of [Margaris] p. 90. See 19.32v 1942 for a version requiring fewer axioms. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)

Hypothesis

Ref	Expression
19.32.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
19.32	⊢ (∀𝑥(𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∀𝑥𝜓))

Proof of Theorem 19.32

Step	Hyp	Ref	Expression
1		19.32.1	. . . 4 ⊢ Ⅎ𝑥𝜑
2	1	nfn 1859	. . 3 ⊢ Ⅎ𝑥 ¬ 𝜑
3	2	19.21 2199	. 2 ⊢ (∀𝑥(¬ 𝜑 → 𝜓) ↔ (¬ 𝜑 → ∀𝑥𝜓))
4		df-or 845	. . 3 ⊢ ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓))
5	4	albii 1820	. 2 ⊢ (∀𝑥(𝜑 ∨ 𝜓) ↔ ∀𝑥(¬ 𝜑 → 𝜓))
6		df-or 845	. 2 ⊢ ((𝜑 ∨ ∀𝑥𝜓) ↔ (¬ 𝜑 → ∀𝑥𝜓))
7	3, 5, 6	3bitr4i 303	1 ⊢ (∀𝑥(𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∀𝑥𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∨ wo 844  ∀wal 1538  Ⅎwnf 1784

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 1912  ax-6 1970  ax-7 2010  ax-12 2170

This theorem depends on definitions:  df-bi 206  df-or 845  df-ex 1781  df-nf 1785

This theorem is referenced by:  19.31  2226  2eu3  2648  axi12  2700  axbnd  2701