Theorem ralnex 2625

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 2620	. 2 ⊢ (∀x ∈ A ¬ φ ↔ ∀x(x ∈ A → ¬ φ))
2		alinexa 1578	. . 3 ⊢ (∀x(x ∈ A → ¬ φ) ↔ ¬ ∃x(x ∈ A ∧ φ))
3		df-rex 2621	. . 3 ⊢ (∃x ∈ A φ ↔ ∃x(x ∈ A ∧ φ))
4	2, 3	xchbinxr 302	. 2 ⊢ (∀x(x ∈ A → ¬ φ) ↔ ¬ ∃x ∈ A φ)
5	1, 4	bitri 240	1 ⊢ (∀x ∈ A ¬ φ ↔ ¬ ∃x ∈ A φ)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   ∈ wcel 1710  ∀wral 2615  ∃wrex 2616

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557

This proof depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-ral 2620  df-rex 2621

This theorem is used by:  dfrex2  2628  ralinexa  2660  nrex  2717  nrexdv  2718  r19.43  2767  rabeq0  3573  iindif2  4036  evenodddisj  4517  rexiunxp  4825