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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

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 1710 ∀wral 2614 ∃wrex 2615

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