Theorem r19.30 2757

Description: Theorem 19.30 of [Margaris] p. 90 with restricted quantifiers. (Contributed by Scott Fenton, 25-Feb-2011.)

Assertion

Ref	Expression
r19.30	⊢ (∀x ∈ A (φ ∨ ψ) → (∀x ∈ A φ ∨ ∃x ∈ A ψ))

Proof of Theorem r19.30

Step	Hyp	Ref	Expression
1		ralim 2686	. 2 ⊢ (∀x ∈ A (¬ ψ → φ) → (∀x ∈ A ¬ ψ → ∀x ∈ A φ))
2		orcom 376	. . . 4 ⊢ ((φ ∨ ψ) ↔ (ψ ∨ φ))
3		df-or 359	. . . 4 ⊢ ((ψ ∨ φ) ↔ (¬ ψ → φ))
4	2, 3	bitri 240	. . 3 ⊢ ((φ ∨ ψ) ↔ (¬ ψ → φ))
5	4	ralbii 2639	. 2 ⊢ (∀x ∈ A (φ ∨ ψ) ↔ ∀x ∈ A (¬ ψ → φ))
6		orcom 376	. . 3 ⊢ ((∀x ∈ A φ ∨ ¬ ∀x ∈ A ¬ ψ) ↔ (¬ ∀x ∈ A ¬ ψ ∨ ∀x ∈ A φ))
7		dfrex2 2628	. . . 4 ⊢ (∃x ∈ A ψ ↔ ¬ ∀x ∈ A ¬ ψ)
8	7	orbi2i 505	. . 3 ⊢ ((∀x ∈ A φ ∨ ∃x ∈ A ψ) ↔ (∀x ∈ A φ ∨ ¬ ∀x ∈ A ¬ ψ))
9		imor 401	. . 3 ⊢ ((∀x ∈ A ¬ ψ → ∀x ∈ A φ) ↔ (¬ ∀x ∈ A ¬ ψ ∨ ∀x ∈ A φ))
10	6, 8, 9	3bitr4i 268	. 2 ⊢ ((∀x ∈ A φ ∨ ∃x ∈ A ψ) ↔ (∀x ∈ A ¬ ψ → ∀x ∈ A φ))
11	1, 5, 10	3imtr4i 257	1 ⊢ (∀x ∈ A (φ ∨ ψ) → (∀x ∈ A φ ∨ ∃x ∈ A ψ))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 357  ∀wral 2615  ∃wrex 2616

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-11 1746

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-ral 2620  df-rex 2621

This theorem is referenced by: (None)