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Mirrors > Home > ILE Home > Th. List > ax-ie2 | GIF version |
Description: Define existential quantification. ∃xφ means "there exists at least one set x such that φ is true." Axiom 10 of 10 for intuitionistic logic. (Contributed by Mario Carneiro, 31-Jan-2015.) |
Ref | Expression |
---|---|
ax-ie2 | ⊢ (∀x(ψ → ∀xψ) → (∀x(φ → ψ) ↔ (∃xφ → ψ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wps | . . . 4 wff ψ | |
2 | vx | . . . . 5 setvar x | |
3 | 1, 2 | wal 1240 | . . . 4 wff ∀xψ |
4 | 1, 3 | wi 4 | . . 3 wff (ψ → ∀xψ) |
5 | 4, 2 | wal 1240 | . 2 wff ∀x(ψ → ∀xψ) |
6 | wph | . . . . 5 wff φ | |
7 | 6, 1 | wi 4 | . . . 4 wff (φ → ψ) |
8 | 7, 2 | wal 1240 | . . 3 wff ∀x(φ → ψ) |
9 | 6, 2 | wex 1378 | . . . 4 wff ∃xφ |
10 | 9, 1 | wi 4 | . . 3 wff (∃xφ → ψ) |
11 | 8, 10 | wb 98 | . 2 wff (∀x(φ → ψ) ↔ (∃xφ → ψ)) |
12 | 5, 11 | wi 4 | 1 wff (∀x(ψ → ∀xψ) → (∀x(φ → ψ) ↔ (∃xφ → ψ))) |
Colors of variables: wff set class |
This axiom is referenced by: 19.23ht 1383 bj-ex 9237 |
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