Theorem rexex 1696

Description: Restricted existence implies existence.

Assertion

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

Proof of Theorem rexex

Step	Hyp	Ref	Expression
1		df-rex 1653	. 2 ⊢ (∃x ∈ A φ ↔ ∃x(x ∈ A ⋀ φ))
2		pm3.27 323	. . 3 ⊢ ((x ∈ A ⋀ φ) → φ)
3	2	19.22i 1042	. 2 ⊢ (∃x(x ∈ A ⋀ φ) → ∃xφ)
4	1, 3	sylbi 199	1 ⊢ (∃x ∈ A φ → ∃xφ)

Syntax hints:   → wi 3   ⋀ wa 223   ∈ wcel 960  ∃wex 982  ∃wrex 1649

This theorem is referenced by:  reu6 1935  dffo5 3827  ivthlem6 7286  ivthlem7 7287

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 965  ax-4 975  ax-5o 977

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-rex 1653

Copyright terms: Public domain