Theorem ralnex 2310

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 2305	. 2 ⊢ (∀x ∈ A ¬ φ ↔ ∀x(x ∈ A → ¬ φ))
2		alinexa 1491	. . 3 ⊢ (∀x(x ∈ A → ¬ φ) ↔ ¬ ∃x(x ∈ A ∧ φ))
3		df-rex 2306	. . 3 ⊢ (∃x ∈ A φ ↔ ∃x(x ∈ A ∧ φ))
4	2, 3	xchbinxr 607	. 2 ⊢ (∀x(x ∈ A → ¬ φ) ↔ ¬ ∃x ∈ A φ)
5	1, 4	bitri 173	1 ⊢ (∀x ∈ A ¬ φ ↔ ¬ ∃x ∈ A φ)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1240  ∃wex 1378   ∈ wcel 1390  ∀wral 2300  ∃wrex 2301

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1333  ax-gen 1335  ax-ie2 1380

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-ral 2305  df-rex 2306

This theorem is referenced by:  rexalim  2313  ralinexa  2345  nrex  2405  nrexdv  2406  uni0b  3596  iindif2m  3715  icc0r  8565