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Mirrors > Home > ILE Home > Th. List > dfexdc | GIF version |
Description: Defining ∃𝑥𝜑 given decidability. It is common in classical logic to define ∃𝑥𝜑 as ¬ ∀𝑥¬ 𝜑 but in intuitionistic logic without a decidability condition, that is only an implication not an equivalence, as seen at exalim 1459. (Contributed by Jim Kingdon, 15-Mar-2018.) |
Ref | Expression |
---|---|
dfexdc | ⊢ (DECID ∃𝑥𝜑 → (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alnex 1456 | . . 3 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑) | |
2 | 1 | a1i 9 | . 2 ⊢ (DECID ∃𝑥𝜑 → (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)) |
3 | 2 | con2biidc 845 | 1 ⊢ (DECID ∃𝑥𝜑 → (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 104 DECID wdc 802 ∀wal 1310 ∃wex 1449 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1404 ax-gen 1406 ax-ie2 1451 |
This theorem depends on definitions: df-bi 116 df-dc 803 df-tru 1315 df-fal 1318 |
This theorem is referenced by: dfrex2dc 2400 |
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