Theorem 19.30 1604

Description: Theorem 19.30 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Assertion

Ref	Expression
19.30	⊢ (∀x(φ ∨ ψ) → (∀xφ ∨ ∃xψ))

Proof of Theorem 19.30

Step	Hyp	Ref	Expression
1		exnal 1574	. . 3 ⊢ (∃x ¬ φ ↔ ¬ ∀xφ)
2		exim 1575	. . 3 ⊢ (∀x(¬ φ → ψ) → (∃x ¬ φ → ∃xψ))
3	1, 2	syl5bir 209	. 2 ⊢ (∀x(¬ φ → ψ) → (¬ ∀xφ → ∃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 ⊢ ((∀xφ ∨ ∃xψ) ↔ (¬ ∀xφ → ∃xψ))
7	3, 5, 6	3imtr4i 257	1 ⊢ (∀x(φ ∨ ψ) → (∀xφ ∨ ∃xψ))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 357  ∀wal 1540  ∃wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557

This theorem depends on definitions:  df-bi 177  df-or 359  df-ex 1542

This theorem is referenced by:  19.33b  1608