Theorem r19.37av 1300

Description: Restricted version of one direction of Theorem 19.37 of [Margaris] p. 90. (The other direction doesn't hold when A is empty.)

Assertion

Ref	Expression
r19.37av	⊢ (∃x ∈ A (φ → ψ) → (φ → ∃x ∈ A ψ))

Distinct variable group(s):   φ,x

Proof of Theorem r19.37av

Step	Hyp	Ref	Expression
1		r19.35 1298	. 2 ⊢ (∃x ∈ A (φ → ψ) ↔ (∀x ∈ A φ → ∃x ∈ A ψ))
2		ax-1 3	. . . 4 ⊢ (φ → (x ∈ A → φ))
3	2	r19.21aiv 1259	. . 3 ⊢ (φ → ∀x ∈ A φ)
4	3	syl4 19	. 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   ∈ wcel 1092  ∀wral 1201  ∃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-gen 677  ax-17 925

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-ral 1205  df-rex 1206

metamath.org