Theorem rspe 2515

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 1578	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		df-rex 2450	. 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 1480   ∈ wcel 2136  ∃wrex 2445

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 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498

This theorem depends on definitions:  df-bi 116  df-rex 2450

This theorem is referenced by:  rsp2e  2517  ssiun2  3909  tfrlem9  6287  tfrlemibxssdm  6295  tfr1onlembxssdm  6311  tfrcllembxssdm  6324  findcard2  6855  findcard2s  6856  prarloclemup  7436  prmuloc2  7508  ltaddpr  7538  aptiprlemu  7581  cauappcvgprlemopl  7587  cauappcvgprlemopu  7589  cauappcvgprlem2  7601  caucvgprlemopl  7610  caucvgprlemopu  7612  caucvgprlem2  7621  caucvgprprlem2  7651  suplocexprlemrl  7658  suplocexprlemru  7660  suplocexprlemlub  7665