Theorem nrex 1270

Description: Inference adding restricted existential quantifier to negated wff.

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 1247	. 2 ⊢ ∀x ∈ A ¬ ψ
3		ralnex 1209	. 2 ⊢ (∀x ∈ A ¬ ψ ↔ ¬ ∃x ∈ A ψ)
4	2, 3	mpbi 164	1 ⊢ ¬ ∃x ∈ A ψ

Syntax hints:  ¬ wn 1   → wi 2   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202

This theorem is referenced by:  rex0 1717  iun0 2028  orduninsuc 2365  cfsuc 3709  nominpos 4509  ruclem37 4921  hatomistic 5755

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-gen 677

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-ral 1205  df-rex 1206

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