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Mirrors > Home > ILE Home > Th. List > rexnalim | GIF version |
Description: Relationship between restricted universal and existential quantifiers. In classical logic this would be a biconditional. (Contributed by Jim Kingdon, 17-Aug-2018.) |
Ref | Expression |
---|---|
rexnalim | ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝜑 → ¬ ∀𝑥 ∈ 𝐴 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rex 2448 | . 2 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ¬ 𝜑)) | |
2 | exanaliim 1634 | . . 3 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ ¬ 𝜑) → ¬ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) | |
3 | df-ral 2447 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) | |
4 | 2, 3 | sylnibr 667 | . 2 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ ¬ 𝜑) → ¬ ∀𝑥 ∈ 𝐴 𝜑) |
5 | 1, 4 | sylbi 120 | 1 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝜑 → ¬ ∀𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∀wal 1340 ∃wex 1479 ∈ wcel 2135 ∀wral 2442 ∃wrex 2443 |
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 604 ax-in2 605 ax-5 1434 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-4 1497 ax-17 1513 ax-ial 1521 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-fal 1348 df-nf 1448 df-ral 2447 df-rex 2448 |
This theorem is referenced by: nnral 2454 ralexim 2456 iundif2ss 3925 ixp0 6688 omniwomnimkv 7122 alzdvds 11777 pc2dvds 12238 nninfsellemeq 13728 |
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