Theorem rspe 2370

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 1482	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		df-rex 2312	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
3	1, 2	sylibr 137	1 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → ∃𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4   ∧ wa 97  ∃wex 1381   ∈ wcel 1393  ∃wrex 2307

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400

This theorem depends on definitions:  df-bi 110  df-rex 2312

This theorem is referenced by:  rsp2e  2372  ssiun2  3700  tfrlem9  5935  tfrlemibxssdm  5941  findcard2  6346  findcard2s  6347  prarloclemup  6593  prmuloc2  6665  ltaddpr  6695  aptiprlemu  6738  cauappcvgprlemopl  6744  cauappcvgprlemopu  6746  cauappcvgprlem2  6758  caucvgprlemopl  6767  caucvgprlemopu  6769  caucvgprlem2  6778  caucvgprprlem2  6808