Theorem r19.44av 2768

Description: One direction of a restricted quantifier version of Theorem 19.44 of [Margaris] p. 90. The other direction doesn't hold when A is empty. (Contributed by NM, 2-Apr-2004.)

Assertion

Ref	Expression
r19.44av	⊢ (∃x ∈ A (φ ∨ ψ) → (∃x ∈ A φ ∨ ψ))

Distinct variable group:   ψ,x

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

Proof of Theorem r19.44av

Step	Hyp	Ref	Expression
1		r19.43 2767	. 2 ⊢ (∃x ∈ A (φ ∨ ψ) ↔ (∃x ∈ A φ ∨ ∃x ∈ A ψ))
2		idd 21	. . . 4 ⊢ (x ∈ A → (ψ → ψ))
3	2	rexlimiv 2733	. . 3 ⊢ (∃x ∈ A ψ → ψ)
4	3	orim2i 504	. 2 ⊢ ((∃x ∈ A φ ∨ ∃x ∈ A ψ) → (∃x ∈ A φ ∨ ψ))
5	1, 4	sylbi 187	1 ⊢ (∃x ∈ A (φ ∨ ψ) → (∃x ∈ A φ ∨ ψ))

Syntax hints:   → wi 4   ∨ wo 357   ∈ wcel 1710  ∃wrex 2616

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 2620  df-rex 2621

This theorem is referenced by: (None)