Theorem rspe 2676

Description: Restricted specialization. (Contributed by NM, 12-Oct-1999.)

Assertion

Ref	Expression
rspe	⊢ ((x ∈ A ∧ φ) → ∃x ∈ A φ)

Proof of Theorem rspe

Step	Hyp	Ref	Expression
1		19.8a 1756	. 2 ⊢ ((x ∈ A ∧ φ) → ∃x(x ∈ A ∧ φ))
2		df-rex 2621	. 2 ⊢ (∃x ∈ A φ ↔ ∃x(x ∈ A ∧ φ))
3	1, 2	sylibr 203	1 ⊢ ((x ∈ A ∧ φ) → ∃x ∈ A φ)

Syntax hints:   → wi 4   ∧ wa 358  ∃wex 1541   ∈ 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-11 1746

This theorem depends on definitions:  df-bi 177  df-ex 1542  df-rex 2621

This theorem is referenced by:  rsp2e  2678  2rmorex  3041  ssiun2  4010  dminss  5042  clos1nrel  5887