Theorem r19.32v 1297

Description: Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers.

Assertion

Ref	Expression
r19.32v	⊢ (∀x ∈ A (φ ∨ ψ) ↔ (φ ∨ ∀x ∈ A ψ))

Distinct variable group(s):   φ,x

Proof of Theorem r19.32v

Step	Hyp	Ref	Expression
1		r19.21v 1260	. 2 ⊢ (∀x ∈ A (¬ φ → ψ) ↔ (¬ φ → ∀x ∈ A ψ))
2		df-or 197	. . 3 ⊢ ((φ ∨ ψ) ↔ (¬ φ → ψ))
3	2	biral 1223	. 2 ⊢ (∀x ∈ A (φ ∨ ψ) ↔ ∀x ∈ A (¬ φ → ψ))
4		df-or 197	. 2 ⊢ ((φ ∨ ∀x ∈ A ψ) ↔ (¬ φ → ∀x ∈ A ψ))
5	1, 3, 4	3bitr4 158	1 ⊢ (∀x ∈ A (φ ∨ ψ) ↔ (φ ∨ ∀x ∈ A ψ))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∨ wo 195  ∀wral 1201

This theorem is referenced by:  iinun2 2031  iinuni 2036

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-6 675  ax-gen 677  ax-17 925

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ral 1205

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