Theorem rexex 2985

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 2902	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		exsimpr 1784	. 2 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ∃𝑥𝜑)
3	1, 2	sylbi 206	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥𝜑)

Syntax hints:   → wi 4   ∧ wa 383  ∃wex 1695   ∈ wcel 1977  ∃wrex 2897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-rex 2902

This theorem is referenced by:  reu3  3363  rmo2i  3493  dffo5  6284  nqerf  9631  supsrlem  9811  vdwmc2  15521  isch3  27482  19.9d2rf  28702  volfiniune  29620  bnj594  30236  bnj1371  30351  bnj1374  30353  dfrdg4  31228  bj-0nelsngl  32152  bj-toprntopon  32244  bj-ccinftydisj  32277  poimirlem25  32604  mblfinlem3  32618  mblfinlem4  32619  clsk3nimkb  37358  stoweidlem57  38950