Theorem 19.31 1089

Description: Theorem 19.31 of [Margaris] p. 90.

Hypothesis

Ref	Expression
19.31.1	⊢ (ψ → ∀xψ)

Assertion

Ref	Expression
19.31	⊢ (∀x(φ ⋁ ψ) ↔ (∀xφ ⋁ ψ))

Proof of Theorem 19.31

Step	Hyp	Ref	Expression
1		19.31.1	. . 3 ⊢ (ψ → ∀xψ)
2	1	19.32 1088	. 2 ⊢ (∀x(ψ ⋁ φ) ↔ (ψ ⋁ ∀xφ))
3		orcom 246	. . 3 ⊢ ((φ ⋁ ψ) ↔ (ψ ⋁ φ))
4	3	albii 1001	. 2 ⊢ (∀x(φ ⋁ ψ) ↔ ∀x(ψ ⋁ φ))
5		orcom 246	. 2 ⊢ ((∀xφ ⋁ ψ) ↔ (ψ ⋁ ∀xφ))
6	2, 4, 5	3bitr4 183	1 ⊢ (∀x(φ ⋁ ψ) ↔ (∀xφ ⋁ ψ))

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

This theorem is referenced by:  19.41 1097  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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