Theorem 19.32 1875

Description: Theorem 19.32 of [Margaris] p. 90. (Contributed by NM, 5-Aug-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 1793	. . 3 ⊢ Ⅎx ¬ φ
3	2	19.21 1796	. 2 ⊢ (∀x(¬ φ → ψ) ↔ (¬ φ → ∀xψ))
4		df-or 359	. . 3 ⊢ ((φ ∨ ψ) ↔ (¬ φ → ψ))
5	4	albii 1566	. 2 ⊢ (∀x(φ ∨ ψ) ↔ ∀x(¬ φ → ψ))
6		df-or 359	. 2 ⊢ ((φ ∨ ∀xψ) ↔ (¬ φ → ∀xψ))
7	3, 5, 6	3bitr4i 268	1 ⊢ (∀x(φ ∨ ψ) ↔ (φ ∨ ∀xψ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357  ∀wal 1540  Ⅎwnf 1544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-or 359  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by:  19.31  1876  2eu3  2286  axi12  2333