Theorem rspe 2546

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 1604	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		df-rex 2481	. 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 1506   ∈ wcel 2167  ∃wrex 2476

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 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524

This theorem depends on definitions:  df-bi 117  df-rex 2481

This theorem is referenced by:  rsp2e  2548  ssiun2  3960  tfrlem9  6386  tfrlemibxssdm  6394  tfr1onlembxssdm  6410  tfrcllembxssdm  6423  findcard2  6959  findcard2s  6960  prarloclemup  7581  prmuloc2  7653  ltaddpr  7683  aptiprlemu  7726  cauappcvgprlemopl  7732  cauappcvgprlemopu  7734  cauappcvgprlem2  7746  caucvgprlemopl  7755  caucvgprlemopu  7757  caucvgprlem2  7766  caucvgprprlem2  7796  suplocexprlemrl  7803  suplocexprlemru  7805  suplocexprlemlub  7810