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Mirrors > Home > NFE Home > Th. List > ralnex | GIF version |
Description: Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997.) |
Ref | Expression |
---|---|
ralnex | ⊢ (∀x ∈ A ¬ φ ↔ ¬ ∃x ∈ A φ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ral 2619 | . 2 ⊢ (∀x ∈ A ¬ φ ↔ ∀x(x ∈ A → ¬ φ)) | |
2 | alinexa 1578 | . . 3 ⊢ (∀x(x ∈ A → ¬ φ) ↔ ¬ ∃x(x ∈ A ∧ φ)) | |
3 | df-rex 2620 | . . 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) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 176 ∧ wa 358 ∀wal 1540 ∃wex 1541 ∈ wcel 1710 ∀wral 2614 ∃wrex 2615 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 |
This theorem depends on definitions: df-bi 177 df-an 360 df-ex 1542 df-ral 2619 df-rex 2620 |
This theorem is referenced by: dfrex2 2627 ralinexa 2659 nrex 2716 nrexdv 2717 r19.43 2766 rabeq0 3572 iindif2 4035 evenodddisj 4516 rexiunxp 4824 |
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