Theorem nrex 2546

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 2507	. 2 ⊢ ∀𝑥 ∈ 𝐴 ¬ 𝜓
3		ralnex 2442	. 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜓 ↔ ¬ ∃𝑥 ∈ 𝐴 𝜓)
4	2, 3	mpbi 144	1 ⊢ ¬ ∃𝑥 ∈ 𝐴 𝜓

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2125  ∀wral 2432  ∃wrex 2433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-5 1424  ax-gen 1426  ax-ie2 1471

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338  df-ral 2437  df-rex 2438

This theorem is referenced by:  rex0  3407  iun0  3901  frec0g  6334  nominpos  9049  sqrt2irr  12008  exmidsbthrlem  13542