Theorem rspe 2519

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 1583	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		df-rex 2454	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
3	1, 2	sylibr 133	1 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → ∃𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4   ∧ wa 103  ∃wex 1485   ∈ wcel 2141  ∃wrex 2449

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503

This theorem depends on definitions:  df-bi 116  df-rex 2454

This theorem is referenced by:  rsp2e  2521  ssiun2  3916  tfrlem9  6298  tfrlemibxssdm  6306  tfr1onlembxssdm  6322  tfrcllembxssdm  6335  findcard2  6867  findcard2s  6868  prarloclemup  7457  prmuloc2  7529  ltaddpr  7559  aptiprlemu  7602  cauappcvgprlemopl  7608  cauappcvgprlemopu  7610  cauappcvgprlem2  7622  caucvgprlemopl  7631  caucvgprlemopu  7633  caucvgprlem2  7642  caucvgprprlem2  7672  suplocexprlemrl  7679  suplocexprlemru  7681  suplocexprlemlub  7686