Theorem rexex 2673

Description: Restricted existence implies existence. (Contributed by NM, 11-Nov-1995.)

Assertion

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

Proof of Theorem rexex

Step	Hyp	Ref	Expression
1		df-rex 2620	. 2 ⊢ (∃x ∈ A φ ↔ ∃x(x ∈ A ∧ φ))
2		simpr 447	. . 3 ⊢ ((x ∈ A ∧ φ) → φ)
3	2	eximi 1576	. 2 ⊢ (∃x(x ∈ A ∧ φ) → ∃xφ)
4	1, 3	sylbi 187	1 ⊢ (∃x ∈ A φ → ∃xφ)

Syntax hints:   → wi 4   ∧ wa 358  ∃wex 1541   ∈ wcel 1710  ∃wrex 2615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-rex 2620

This theorem is referenced by:  reu3  3026  rmo2i  3132  dffo5  5424