Theorem rexex 1242

Description: Restricted existence implies existence.

Assertion

Ref	Expression
rexex	⊢ (∃x ∈ A φ → ∃xφ)

Proof of Theorem rexex

Step	Hyp	Ref	Expression
1		df-rex 1206	. 2 ⊢ (∃x ∈ A φ ↔ ∃x(x ∈ A ∧ φ))
2		pm3.27 260	. . 3 ⊢ ((x ∈ A ∧ φ) → φ)
3	2	19.22i 723	. 2 ⊢ (∃x(x ∈ A ∧ φ) → ∃xφ)
4	1, 3	sylbi 174	1 ⊢ (∃x ∈ A φ → ∃xφ)

Syntax hints:   → wi 2   ∧ wa 196  ∃wex 678   ∈ 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-gen 677

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

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