Theorem rspe 2591

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 1639	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		df-rex 2526	. 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 1541   ∈ wcel 2203  ∃wrex 2521

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 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559

This theorem depends on definitions:  df-bi 117  df-rex 2526

This theorem is referenced by:  rsp2e  2593  ssiun2  4033  tfrlem9  6549  tfrlemibxssdm  6557  tfr1onlembxssdm  6573  tfrcllembxssdm  6586  findcard2  7145  findcard2s  7146  prarloclemup  7806  prmuloc2  7878  ltaddpr  7908  aptiprlemu  7951  cauappcvgprlemopl  7957  cauappcvgprlemopu  7959  cauappcvgprlem2  7971  caucvgprlemopl  7980  caucvgprlemopu  7982  caucvgprlem2  7991  caucvgprprlem2  8021  suplocexprlemrl  8028  suplocexprlemru  8030  suplocexprlemlub  8035