Theorem rspe 2559

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 1616	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		df-rex 2494	. 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 1518   ∈ wcel 2180  ∃wrex 2489

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 1475  ax-ie1 1519  ax-ie2 1520  ax-4 1536

This theorem depends on definitions:  df-bi 117  df-rex 2494

This theorem is referenced by:  rsp2e  2561  ssiun2  3987  tfrlem9  6435  tfrlemibxssdm  6443  tfr1onlembxssdm  6459  tfrcllembxssdm  6472  findcard2  7019  findcard2s  7020  prarloclemup  7650  prmuloc2  7722  ltaddpr  7752  aptiprlemu  7795  cauappcvgprlemopl  7801  cauappcvgprlemopu  7803  cauappcvgprlem2  7815  caucvgprlemopl  7824  caucvgprlemopu  7826  caucvgprlem2  7835  caucvgprprlem2  7865  suplocexprlemrl  7872  suplocexprlemru  7874  suplocexprlemlub  7879