Theorem ralnex 2493

Description: Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997.)

Assertion

Ref	Expression
ralnex	⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 𝜑)

Proof of Theorem ralnex

Step	Hyp	Ref	Expression
1		df-ral 2488	. 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝜑))
2		alinexa 1625	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝜑) ↔ ¬ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
3		df-rex 2489	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
4	2, 3	xchbinxr 684	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝜑) ↔ ¬ ∃𝑥 ∈ 𝐴 𝜑)
5	1, 4	bitri 184	1 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1370  ∃wex 1514   ∈ wcel 2175  ∀wral 2483  ∃wrex 2484

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 615  ax-in2 616  ax-5 1469  ax-gen 1471  ax-ie2 1516

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-fal 1378  df-ral 2488  df-rex 2489

This theorem is referenced by:  nnral  2495  rexalim  2498  ralinexa  2532  nrex  2597  nrexdv  2598  ralnex2  2644  r19.30dc  2652  uni0b  3874  iindif2m  3994  f0rn0  5469  supmoti  7094  fodjuomnilemdc  7245  ismkvnex  7256  nninfwlpoimlemginf  7277  suprnubex  9025  icc0r  10047  ioo0  10400  ico0  10402  ioc0  10403  prmind2  12413  sqrt2irr  12455  nconstwlpolem  15966