Theorem rspe 2579

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 1636	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		df-rex 2514	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
3	1, 2	sylibr 134	1 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → ∃𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4   ∧ wa 104  ∃wex 1538   ∈ wcel 2200  ∃wrex 2509

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556

This theorem depends on definitions:  df-bi 117  df-rex 2514

This theorem is referenced by:  rsp2e  2581  ssiun2  4008  tfrlem9  6471  tfrlemibxssdm  6479  tfr1onlembxssdm  6495  tfrcllembxssdm  6508  findcard2  7059  findcard2s  7060  prarloclemup  7690  prmuloc2  7762  ltaddpr  7792  aptiprlemu  7835  cauappcvgprlemopl  7841  cauappcvgprlemopu  7843  cauappcvgprlem2  7855  caucvgprlemopl  7864  caucvgprlemopu  7866  caucvgprlem2  7875  caucvgprprlem2  7905  suplocexprlemrl  7912  suplocexprlemru  7914  suplocexprlemlub  7919