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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 2460 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝜑)) | |
2 | alinexa 1603 | . . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝜑) ↔ ¬ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
3 | df-rex 2461 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
4 | 2, 3 | xchbinxr 683 | . 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝜑) ↔ ¬ ∃𝑥 ∈ 𝐴 𝜑) |
5 | 1, 4 | bitri 184 | 1 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∀wal 1351 ∃wex 1492 ∈ wcel 2148 ∀wral 2455 ∃wrex 2456 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-5 1447 ax-gen 1449 ax-ie2 1494 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-fal 1359 df-ral 2460 df-rex 2461 |
This theorem is referenced by: nnral 2467 rexalim 2470 ralinexa 2504 nrex 2569 nrexdv 2570 ralnex2 2616 r19.30dc 2624 uni0b 3836 iindif2m 3956 f0rn0 5412 supmoti 6995 fodjuomnilemdc 7145 ismkvnex 7156 nninfwlpoimlemginf 7177 suprnubex 8913 icc0r 9929 ioo0 10263 ico0 10265 ioc0 10266 prmind2 12123 sqrt2irr 12165 nconstwlpolem 14954 |
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