Theorem r19.45av 1306

Description: Restricted version of one direction of Theorem 19.45 of [Margaris] p. 90. (The other direction doesn't hold when A is empty.)

Assertion

Ref	Expression
r19.45av	⊢ (∃x ∈ A (φ ∨ ψ) → (φ ∨ ∃x ∈ A ψ))

Distinct variable group(s):   φ,x

Proof of Theorem r19.45av

Step	Hyp	Ref	Expression
1		r19.43 1304	. 2 ⊢ (∃x ∈ A (φ ∨ ψ) ↔ (∃x ∈ A φ ∨ ∃x ∈ A ψ))
2		idd 11	. . . 4 ⊢ (x ∈ A → (φ → φ))
3	2	r19.23aiv 1284	. . 3 ⊢ (∃x ∈ A φ → φ)
4	3	orim1i 272	. 2 ⊢ ((∃x ∈ A φ ∨ ∃x ∈ A ψ) → (φ ∨ ∃x ∈ A ψ))
5	1, 4	sylbi 174	1 ⊢ (∃x ∈ A (φ ∨ ψ) → (φ ∨ ∃x ∈ A ψ))

Syntax hints:   → wi 2   ∨ wo 195   ∈ wcel 1092  ∃wrex 1202

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-ex 679  df-rex 1206

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