Theorem nrex 2602

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 2563	. 2 ⊢ ∀𝑥 ∈ 𝐴 ¬ 𝜓
3		ralnex 2498	. 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜓 ↔ ¬ ∃𝑥 ∈ 𝐴 𝜓)
4	2, 3	mpbi 145	1 ⊢ ¬ ∃𝑥 ∈ 𝐴 𝜓

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2180  ∀wral 2488  ∃wrex 2489

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-5 1473  ax-gen 1475  ax-ie2 1520

This theorem depends on definitions:  df-bi 117  df-tru 1378  df-fal 1381  df-ral 2493  df-rex 2494

This theorem is referenced by:  rex0  3489  iun0  4001  canth  5925  frec0g  6513  nominpos  9317  sqrt2irr  12650  gsum0g  13395  exmidsbthrlem  16301