Theorem rspe 2526

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 1590	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		df-rex 2461	. 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 1492   ∈ wcel 2148  ∃wrex 2456

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 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510

This theorem depends on definitions:  df-bi 117  df-rex 2461

This theorem is referenced by:  rsp2e  2528  ssiun2  3930  tfrlem9  6320  tfrlemibxssdm  6328  tfr1onlembxssdm  6344  tfrcllembxssdm  6357  findcard2  6889  findcard2s  6890  prarloclemup  7494  prmuloc2  7566  ltaddpr  7596  aptiprlemu  7639  cauappcvgprlemopl  7645  cauappcvgprlemopu  7647  cauappcvgprlem2  7659  caucvgprlemopl  7668  caucvgprlemopu  7670  caucvgprlem2  7679  caucvgprprlem2  7709  suplocexprlemrl  7716  suplocexprlemru  7718  suplocexprlemlub  7723