Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE 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 2453 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝜑)) | |
2 | alinexa 1596 | . . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝜑) ↔ ¬ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
3 | df-rex 2454 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
4 | 2, 3 | xchbinxr 678 | . 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝜑) ↔ ¬ ∃𝑥 ∈ 𝐴 𝜑) |
5 | 1, 4 | bitri 183 | 1 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∀wal 1346 ∃wex 1485 ∈ wcel 2141 ∀wral 2448 ∃wrex 2449 |
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 609 ax-in2 610 ax-5 1440 ax-gen 1442 ax-ie2 1487 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-fal 1354 df-ral 2453 df-rex 2454 |
This theorem is referenced by: nnral 2460 rexalim 2463 ralinexa 2497 nrex 2562 nrexdv 2563 ralnex2 2609 r19.30dc 2617 uni0b 3821 iindif2m 3940 f0rn0 5392 supmoti 6970 fodjuomnilemdc 7120 ismkvnex 7131 nninfwlpoimlemginf 7152 suprnubex 8869 icc0r 9883 ioo0 10216 ico0 10218 ioc0 10219 prmind2 12074 sqrt2irr 12116 nconstwlpolem 14096 |
Copyright terms: Public domain | W3C validator |