Theorem nrex 3075

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 3064	. 2 ⊢ ∀𝑥 ∈ 𝐴 ¬ 𝜓
3		ralnex 3073	. 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜓 ↔ ¬ ∃𝑥 ∈ 𝐴 𝜓)
4	2, 3	mpbi 229	1 ⊢ ¬ ∃𝑥 ∈ 𝐴 𝜓

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2107  ∀wral 3062  ∃wrex 3071

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812

This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-ral 3063  df-rex 3072

This theorem is referenced by:  rex0  4358  iun0  5066  canth  7362  orduninsuc  7832  wfrlem16OLD  8324  wofib  9540  cfsuc  10252  nominpos  12449  nnunb  12468  indstr  12900  eirr  16148  sqrt2irr  16192  vdwap0  16909  smndex1n0mnd  18793  smndex2dnrinv  18796  psgnunilem3  19364  bwth  22914  zfbas  23400  aaliou3lem9  25863  vma1  26670  muls01  27568  mulsrid  27569  hatomistici  31615  esumrnmpt2  33066  fmlan0  34382  linedegen  35115  limsucncmpi  35330  elpadd0  38680  rexanuz2nf  44203  fourierdlem62  44884  etransc  44999  0nodd  46580  2nodd  46582  1neven  46830