Theorem rexex 2576

Description: Restricted existence implies existence. (Contributed by NM, 11-Nov-1995.)

Assertion

Ref	Expression
rexex	⊢ (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥𝜑)

Proof of Theorem rexex

Step	Hyp	Ref	Expression
1		df-rex 2514	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		simpr 110	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜑)
3	2	eximi 1646	. 2 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ∃𝑥𝜑)
4	1, 3	sylbi 121	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥𝜑)

Syntax hints:   → wi 4   ∧ wa 104  ∃wex 1538   ∈ wcel 2200  ∃wrex 2509

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-5 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-ial 1580

This theorem depends on definitions:  df-bi 117  df-rex 2514

This theorem is referenced by:  reu3  2993  rmo2i  3120  dffo5  5777  halfnq  7586  nsmallnq  7588  0npr  7658  genpml  7692  genpmu  7693  ltexprlemm  7775  ltexprlemloc  7782  dedekindeulemlub  15279  dedekindicclemlub  15288