Theorem nrex 2717

Description: Inference adding restricted existential quantifier to negated wff. (Contributed by NM, 16-Oct-2003.)

Hypothesis

Ref	Expression
nrex.1	⊢ (x ∈ A → ¬ ψ)

Assertion

Ref	Expression
nrex	⊢ ¬ ∃x ∈ A ψ

Proof of Theorem nrex

Step	Hyp	Ref	Expression
1		nrex.1	. . 3 ⊢ (x ∈ A → ¬ ψ)
2	1	rgen 2680	. 2 ⊢ ∀x ∈ A ¬ ψ
3		ralnex 2625	. 2 ⊢ (∀x ∈ A ¬ ψ ↔ ¬ ∃x ∈ A ψ)
4	2, 3	mpbi 199	1 ⊢ ¬ ∃x ∈ A ψ

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 1710  ∀wral 2615  ∃wrex 2616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-ral 2620  df-rex 2621

This theorem is referenced by:  rex0  3564  iun0  4023  0nelsuc  4401  addcnul1  4453  nulnnn  4557  0cnelphi  4598  proj1op  4601  proj2op  4602  nenpw1pwlem2  6086  nchoice  6309  fnfreclem2  6319