Theorem rspe 2543

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 1601	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		df-rex 2478	. 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 1503   ∈ wcel 2164  ∃wrex 2473

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 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521

This theorem depends on definitions:  df-bi 117  df-rex 2478

This theorem is referenced by:  rsp2e  2545  ssiun2  3955  tfrlem9  6372  tfrlemibxssdm  6380  tfr1onlembxssdm  6396  tfrcllembxssdm  6409  findcard2  6945  findcard2s  6946  prarloclemup  7555  prmuloc2  7627  ltaddpr  7657  aptiprlemu  7700  cauappcvgprlemopl  7706  cauappcvgprlemopu  7708  cauappcvgprlem2  7720  caucvgprlemopl  7729  caucvgprlemopu  7731  caucvgprlem2  7740  caucvgprprlem2  7770  suplocexprlemrl  7777  suplocexprlemru  7779  suplocexprlemlub  7784