Theorem rexex 2482

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 2423	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		simpr 109	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜑)
3	2	eximi 1580	. 2 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ∃𝑥𝜑)
4	1, 3	sylbi 120	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥𝜑)

Syntax hints:   → wi 4   ∧ wa 103  ∃wex 1469   ∈ wcel 1481  ∃wrex 2418

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 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-ial 1515

This theorem depends on definitions:  df-bi 116  df-rex 2423

This theorem is referenced by:  reu3  2878  rmo2i  3003  dffo5  5577  halfnq  7243  nsmallnq  7245  0npr  7315  genpml  7349  genpmu  7350  ltexprlemm  7432  ltexprlemloc  7439  dedekindeulemlub  12806  dedekindicclemlub  12815