Theorem 19.30 764

Description: Theorem 19.30 of [Margaris] p. 90.

Assertion

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

Proof of Theorem 19.30

Step	Hyp	Ref	Expression
1		19.20 690	. 2 ⊢ (∀x(¬ ψ → φ) → (∀x ¬ ψ → ∀xφ))
2		orcom 209	. . . 4 ⊢ ((φ ∨ ψ) ↔ (ψ ∨ φ))
3		df-or 197	. . . 4 ⊢ ((ψ ∨ φ) ↔ (¬ ψ → φ))
4	2, 3	bitr 151	. . 3 ⊢ ((φ ∨ ψ) ↔ (¬ ψ → φ))
5	4	bial 695	. 2 ⊢ (∀x(φ ∨ ψ) ↔ ∀x(¬ ψ → φ))
6		orcom 209	. . 3 ⊢ ((∀xφ ∨ ¬ ∀x ¬ ψ) ↔ (¬ ∀x ¬ ψ ∨ ∀xφ))
7		df-ex 679	. . . 4 ⊢ (∃xψ ↔ ¬ ∀x ¬ ψ)
8	7	orbi2i 214	. . 3 ⊢ ((∀xφ ∨ ∃xψ) ↔ (∀xφ ∨ ¬ ∀x ¬ ψ))
9		imor 204	. . 3 ⊢ ((∀x ¬ ψ → ∀xφ) ↔ (¬ ∀x ¬ ψ ∨ ∀xφ))
10	6, 8, 9	3bitr4 158	. 2 ⊢ ((∀xφ ∨ ∃xψ) ↔ (∀x ¬ ψ → ∀xφ))
11	1, 5, 10	3imtr4 192	1 ⊢ (∀x(φ ∨ ψ) → (∀xφ ∨ ∃xψ))

Syntax hints:  ¬ wn 1   → wi 2   ∨ wo 195  ∀wal 672  ∃wex 678

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-gen 677

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679

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