Theorem rspe 2526

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 1590	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		df-rex 2461	. 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 1492   ∈ wcel 2148  ∃wrex 2456

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 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510

This theorem depends on definitions:  df-bi 117  df-rex 2461

This theorem is referenced by:  rsp2e  2528  ssiun2  3929  tfrlem9  6319  tfrlemibxssdm  6327  tfr1onlembxssdm  6343  tfrcllembxssdm  6356  findcard2  6888  findcard2s  6889  prarloclemup  7493  prmuloc2  7565  ltaddpr  7595  aptiprlemu  7638  cauappcvgprlemopl  7644  cauappcvgprlemopu  7646  cauappcvgprlem2  7658  caucvgprlemopl  7667  caucvgprlemopu  7669  caucvgprlem2  7678  caucvgprprlem2  7708  suplocexprlemrl  7715  suplocexprlemru  7717  suplocexprlemlub  7722