Theorem 19.32 1088

Description: Theorem 19.32 of [Margaris] p. 90.

Hypothesis

Ref	Expression
19.32.1	⊢ (φ → ∀xφ)

Assertion

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

Proof of Theorem 19.32

Step	Hyp	Ref	Expression
1		19.32.1	. . . 4 ⊢ (φ → ∀xφ)
2	1	hbn 1006	. . 3 ⊢ (¬ φ → ∀x ¬ φ)
3	2	19.21 1058	. 2 ⊢ (∀x(¬ φ → ψ) ↔ (¬ φ → ∀xψ))
4		df-or 224	. . 3 ⊢ ((φ ⋁ ψ) ↔ (¬ φ → ψ))
5	4	albii 1001	. 2 ⊢ (∀x(φ ⋁ ψ) ↔ ∀x(¬ φ → ψ))
6		df-or 224	. 2 ⊢ ((φ ⋁ ∀xψ) ↔ (¬ φ → ∀xψ))
7	3, 5, 6	3bitr4 183	1 ⊢ (∀x(φ ⋁ ψ) ↔ (φ ⋁ ∀xψ))

Syntax hints:  ¬ wn 2   → wi 3   ↔ wb 146   ⋁ wo 222  ∀wal 956

This theorem is referenced by:  19.31 1089  2eu3 1454

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 965  ax-4 975  ax-5o 977  ax-6o 980

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225

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