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  4011  tfrlem9  6480  tfrlemibxssdm  6488  tfr1onlembxssdm  6504  tfrcllembxssdm  6517  findcard2  7073  findcard2s  7074  prarloclemup  7708  prmuloc2  7780  ltaddpr  7810  aptiprlemu  7853  cauappcvgprlemopl  7859  cauappcvgprlemopu  7861  cauappcvgprlem2  7873  caucvgprlemopl  7882  caucvgprlemopu  7884  caucvgprlem2  7893  caucvgprprlem2  7923  suplocexprlemrl  7930  suplocexprlemru  7932  suplocexprlemlub  7937