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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 2353 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝜑)) | |
2 | alinexa 1534 | . . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝜑) ↔ ¬ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
3 | df-rex 2354 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
4 | 2, 3 | xchbinxr 640 | . 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝜑) ↔ ¬ ∃𝑥 ∈ 𝐴 𝜑) |
5 | 1, 4 | bitri 182 | 1 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 102 ↔ wb 103 ∀wal 1282 ∃wex 1421 ∈ wcel 1433 ∀wral 2348 ∃wrex 2349 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-5 1376 ax-gen 1378 ax-ie2 1423 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-fal 1290 df-ral 2353 df-rex 2354 |
This theorem is referenced by: rexalim 2361 ralinexa 2393 nrex 2453 nrexdv 2454 uni0b 3626 iindif2m 3745 supmoti 6406 suprnubex 8031 icc0r 8949 ioo0 9268 ico0 9270 ioc0 9271 prmind2 10502 sqrt2irr 10541 |
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