Theorem r19.32v 2757

Description: Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 25-Nov-2003.)

Assertion

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

Distinct variable group:   φ,x

Allowed substitution hints:   ψ(x)   A(x)

Proof of Theorem r19.32v

Step	Hyp	Ref	Expression
1		r19.21v 2701	. 2 ⊢ (∀x ∈ A (¬ φ → ψ) ↔ (¬ φ → ∀x ∈ A ψ))
2		df-or 359	. . 3 ⊢ ((φ ∨ ψ) ↔ (¬ φ → ψ))
3	2	ralbii 2638	. 2 ⊢ (∀x ∈ A (φ ∨ ψ) ↔ ∀x ∈ A (¬ φ → ψ))
4		df-or 359	. 2 ⊢ ((φ ∨ ∀x ∈ A ψ) ↔ (¬ φ → ∀x ∈ A ψ))
5	1, 3, 4	3bitr4i 268	1 ⊢ (∀x ∈ A (φ ∨ ψ) ↔ (φ ∨ ∀x ∈ A ψ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357  ∀wral 2614

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-6 1729  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 2619

This theorem is referenced by:  iinun2  4032  iinuni  4049