Theorem rexex 2516

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 2454	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		simpr 109	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜑)
3	2	eximi 1593	. 2 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ∃𝑥𝜑)
4	1, 3	sylbi 120	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥𝜑)

Syntax hints:   → wi 4   ∧ wa 103  ∃wex 1485   ∈ wcel 2141  ∃wrex 2449

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-rex 2454

This theorem is referenced by:  reu3  2920  rmo2i  3045  dffo5  5645  halfnq  7373  nsmallnq  7375  0npr  7445  genpml  7479  genpmu  7480  ltexprlemm  7562  ltexprlemloc  7569  dedekindeulemlub  13392  dedekindicclemlub  13401