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  7579  prmuloc2  7651  ltaddpr  7681  aptiprlemu  7724  cauappcvgprlemopl  7730  cauappcvgprlemopu  7732  cauappcvgprlem2  7744  caucvgprlemopl  7753  caucvgprlemopu  7755  caucvgprlem2  7764  caucvgprprlem2  7794  suplocexprlemrl  7801  suplocexprlemru  7803  suplocexprlemlub  7808