Theorem nrex 2716

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 2679	. 2 ⊢ ∀x ∈ A ¬ ψ
3		ralnex 2624	. 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 2614  ∃wrex 2615

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 2619  df-rex 2620

This theorem is referenced by:  rex0  3563  iun0  4022  0nelsuc  4400  addcnul1  4452  nulnnn  4556  0cnelphi  4597  proj1op  4600  proj2op  4601  nenpw1pwlem2  6085  nchoice  6308  fnfreclem2  6318